In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

chapter 1 sets exercise 1.6 cbse ncert 11th mathematics

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

C = set of people who like coffee

T =set of people who like tea

n(C)=37

n(T) =52

 because it is given that each person likes at least one of the two drinks

n(C union T)=70 

n(C union T) = n(C) +n(T) - n(C intersection T)

70 = 37 +52 -n(C intersection T)

n(C intersection T) = 37+52 -70 =19

 

19 persons like both coffee and tea.

 

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

F = set of people who speak french

S= set of people who speak spanish

n(F)=50

n(S)=20

n( F intersection S ) =10

n( F union S ) = n(F) +n(S) - n( F intersection S )

= 50 +20 -10 =60

60 people speak at least one of these two languages.

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution 

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution

 

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

 

In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B an

16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

 

A =set of people who liked product A

B =set of people who liked product B

C =set of people who liked product C

 

n(A) =21

n(B)=26

n(C)=29

 

n( A intersection B )=14

n( B intersection C )=14

n( C intersection A )=12

n( A intersection B intersection C )=8

 

removing,  n( A intersection B intersection C )=8

 

n( B intersection C and not A )=14-8 = 6

n( C intersection A and not B )=12-8 =4

We remove these two and also n( A intersection B intersection C )=8

to get 

n(C)  - n( B intersection C and not A ) - n( C intersection A and not B ) -n( A intersection B intersection C )

=29-6-4-8=11 

therefore

11 people liked only product C

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

 

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find: (i) the number of people who read at least one of the newspapers. (ii) the number of people who read exactly one newspaper

ncert cbse 11th mathematics chapter 1 sets miscellaneous exercise

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

 

H = set of people who read H

T  = set of people who read T

 I = set of people who read I

 

n(H)  = 25

n(T) = 26

n(I) =26

n( H intersection I ) = 9

n( H intersection T ) = 11

 n( T intersection I ) = 8

 n( H intersection T intersection I ) = 3

 

i)

at least one of the newspapers means ( H union T union I )

 

 n( H union T union I ) = n(H) + n(T)+n(I) - n( H intersection I ) -n( H intersection T ) -n( T intersection I ) +  n( H intersection T intersection I )

 

n( H union T union I ) =  25+26+26-9-11-8+3 =52

therefore 52 people read at least one of the newspapers.

 

(ii)

 

let n(H and I only and not T) = a

n(H and T only and not I) = b

n(T and I only and not H) = c

 

using n( H intersection T intersection I ) = 3

and given values of n( H intersection I ), n( H intersection T ),( T intersection I )

 

n( H intersection I ) =>   a+3 =   9

n( H intersection T ) =>  b+3 = 11

 n( T intersection I ) =>   c+3 =  8

 ----------------------------------------------------------------

adding

                          a + b + c + 9 =28

 

so a+b+c = 28-9

a+b+c = 19 

19 people read exactly two of the newspapers

 

now we add the people who read exactly three of the newspapers namely

adding n( H intersection T intersection I ) = 3 on both sides

a+b+c +n( H intersection T intersection I ) =19+3 = 22

22 people read more than one newspaper

remove these 22 people to get

 

n( H union T union I ) - [a+b+c +n( H intersection T intersection I )]

= 52 - 22 = 30

the number of people who read exactly one newspaper = 30

 

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

 

In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

 chapter 1 miscellaneous sets ncert cbse

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

 

let A =set of students taking tea

B = set of students taking coffee

U = universal set 

n(U) =600

n(A)  = 150

n(B) =225

n(A intersection B) =100

n(A union B) = n(A) + n(B) - n(A intersection B) 

                    = 150 + 225 -100 =275

 

number of students were taking neither tea nor coffee

= n(U)  - n(A union B)

= 600 -275

=325 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

let A = set of students who know hindi

B = set of students who know english

n(A) =100

n(B) = 50

n(A intersection B) =25

 

n(A union B) = n(A) + n(B) - n(A intersection B) 

 =100 + 50 -25 = 125

 =================================================

 

 chapter 1 miscellaneous sets ncert cbse

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

 ncert cbse 11th chapter 2 relations and functions miscellaneous exercise

 

8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 

(1,1)  is in f 

 

so

x=1, y=1 or f(1)=1 means 

f(1) = a(1) + b can be rewritten as 1= a + b  or a+b=1

 

(2,3) is in f

so 

x=2 , y = 3 or f(2)  = 3 

means

f(2) = a(2) + b can be re written as 3 = 2a+b or 2a+b =3

  a+b=1

2a+b =3

solve by elimination method

subtracting gives 

a = 2 

resubstitute to get b = (-1) 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 

(i) if b =2, 

a=(b^2) =(2^2) = 4

so only (4,2)∈ R and (2,2) is not in R [ because 2^2 is not 2 ]

OR assume  (2,2)∈ R

a=2 , b=2 so that a=(b^2)  is true or 2 =(2^2) which is clearly false

so (2,2) is not in R

 

(a,a) ∈ R, for all a ∈ N is FALSE.

(ii)

if b =2, 

a=(b^2) =(2^2) = 4

so (4,2)∈ R

but if b=4 ,then a =b^2 = 4^2 =16

so that only (16,4) ∈ R and (2,4) is not in R [ because 4^2 is not 2 ]

 (a,b) ∈ R, implies (b,a) ∈ R is FALSE.

(iii)

if b=4 ,then a =b^2 = 4^2 =16

so that  (16,4) ∈ R

if b =2, 

a=(b^2) =(2^2) = 4

so (4,2)∈ R

now both

 (16,4) ∈ R and (4,2)∈ R

but (16,2)  is not in R because (2^2) is not 16

 (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R is FALSE

 

=================================================

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution 

Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

 11th cbse ncert chapter 2 relations and functions miscellaneous exercise

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

9 = 3*3 therefore f(9)=3  

10= 2*5 therefore f(10)=5

11 =11 therefore f(11)=11

12 = 2 * 2 * 3 therefore f(12)=3

13 =13 therefore f(13)=13  

range of f = {3,5,11,13}

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to Z? Justify your answer.

choose a =(-1), b=(-1)

a*b = (-1)*(-1)  = 1

a+b = (-1)+(-1) = (-2)

therefore ( 1, -2 ) belongs to f

now choose a =1 , b =1

a*b = 1*1 = 1

a+b = 1+1 =2

therefore (1,2) belongs to f

therefore the element 1 has more than one image under f namely 

(-2) and 2.

Therefore f is not a function .

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case. 

clearly  each element of f belongs to A x B. So f is a relation on A to B

both (2,9) and (2,11) belong to f 

so f has more than one image under f

so f is not a function.

 

=================================================

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to Z? Justify your answer.

solution

 

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution

 

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

solution  

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution  

 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

solution

Find the area of the triangle formed by the lines joining the vertex of the parabola
x^2 = 12y to the ends of its latus rectum.  

solution 

7. A man running a racecourse notes that the sum of the distances from the two flagposts from him is always 10 m and the distance between the flag posts is 8 m.
Find the equation of the posts traced by the man

solution 

8. An equilateral triangle is inscribed in the parabola (y^2) = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

 solution 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man

 ncert cbse 11th mathematics chapter 11 conic section

 

7. A man running a racecourse notes that the sum of the distances from the two flagposts from him is always 10 m and the distance between the flag posts is 8 m.
Find the equation of the posts traced by the man

let P be a point on the locus

If we take the flagposts as F and F'

we are given that PF + PF' =10m ( a constant )

FF' = 8m

Using properties of an ellipse , the locus is an ellipse with

foci at F and F'

and PF + PF' =2a , and FF' = 2c

so that 2a = 10  and 2c = 8

or a = 10/2 = 5 , so  a^2 = 25

c = 8/2 = 4

use (c^2) = (a^2) - (b^2)

(b^2) = (a^2) - (c^2)

(b^2) = (5^2) - (4^2) =25 -16 = 9

 (b^2) =9 note that this is (b^2) not just b

equation of ellipse is 

[(x^2) / (a^2)] + [(y^2) / (b^2)] = 1

 [(x^2) / (25)] + [(y^2) / (9)] = 1

 

8. An equilateral triangle is inscribed in the parabola (y^2) = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

 Let u be the side of the equilateral triangle

Given the vertex of the parabola O(0,0)  is also a vertex of the equilateral triangle

 If A and B are the other two vertices,  the axis of the parabola, x axis is a perpendicular bisector of AB.

let M be the point of intersection of AB and the x axis.

clearly OA = AB =OB = u

and BM =MA =(u/2) 

By symmetry of the parabola about the x axis, we can choose

the vertices A and  B as A( x,(u/2) ) and B( x,(-u/2) )

clearly MO = x

MA = (u/2) 

using pythagoras theorem in right angled triangle OAM

(x^2) = (u^2) - [(u/2)^2]

(x^2) = (u^2) - [(u^2)/ 4 ]

 (x^2) =[3(u^2)/ 4 ]

x = sqrt(3) * u /2

so that A( x,(u/2) ) changes to A( {sqrt(3) * u /2} , (u/2)  )

A lies on  (y^2) = 4 ax,

so  (u/2)^2 = 4a {sqrt(3) * u /2}

  (u^2) / 4 = 2a *sqrt(3)*u

since u cannot be zero

u = 8 * sqrt(3) *a

 

 

 

=================================================

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution

 

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

solution  

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution  

 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

solution

Find the area of the triangle formed by the lines joining the vertex of the parabola
x^2 = 12y to the ends of its latus rectum.  

solution 

7. A man running a racecourse notes that the sum of the distances from the two flagposts from him is always 10 m and the distance between the flag posts is 8 m.
Find the equation of the posts traced by the man

solution 

8. An equilateral triangle is inscribed in the parabola (y^2) = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

 solution 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

 

let the rod touch the x axis at A and y axis at B with origin at O

given PA=3

so PB=12-3 = 9

Draw PM perpendicular to the x axis and

PN perpendicular to the y axis

so if P=(x,y) 

PM=y and PN=x

 

Clearly if the rod makes an angle u with the x axis

ie. , if  angle(OAP) = u

then angle(NPB) = u

using triangle APM 

sin u = (y/3)

using triangle BPN

cos u = (x/9)

 

We know that 

[cos u] ^2  + [sin u]^2  = 1

so

(x/9)^2 + (y/3)^2 = 1

[(x^2) / 81]  + [(y^2) / 9] =1

 

Find the area of the triangle formed by the lines joining the vertex of the parabola
x^2 = 12y to the ends of its latus rectum. 

use x^2 = 4ay with x^2 = 12y

to get 4a = 12

or a=12/4 

a=3 

Focus is (0,a)  = (0,3)

so latus rectum = 4a =4*3 = 12

semi latus rectum = 12 / 2 = 6

 

end points of the latus rectum are (6,3)  and (-6,3)

vertex is (0,0)

Now we can either use the formula for area using these three coordinates

given by

 area of a  triangle =

{1/2}[x1(y2-y3) + x2(y3-y1) +x3(y1-y2)]

 

 

OR

 

use the fact that

vertex V=(0,0) ,focus F=(0,3)

height is along the axis VF = 3units

base is along the latus rectum of length =12

so area = (1/2) *base *height

=(1/2)*12*3 =18 square units.

 

=================================================

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution

 

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

solution  

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution  

 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

solution

Find the area of the triangle formed by the lines joining the vertex of the parabola
x^2 = 12y to the ends of its latus rectum.  

solution 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.

ncert cbse 11th mathematics  conic sections miscellaneous

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

Chose the parabola as (x^2)  = 4ay (open upwards)

Because the shortest wire is 6m at the centre , the road is hanging 6m below the vertex of the parabola, 

so the 6m has to be subtracted for all data to get distance above x axis on the parabola

 100 m long road means 50m either side of vertex 

longest wire 30 m will at either end 

and 30-6 = 24 m will be above x axis

so (50,24) lies on (x^2)  = 4ay

so 50^2 = 4 a (24) 

a = 2500 / 96

(x^2)  = 4ay

changes to  (x^2)  = 4(2500/96)y 

at 18m use (18,y)  which lies  on (x^2)  = 4(2500/96)y 

so 18^2 = 4(2500/96)y 

y =96*(18^2) / [4*2500]

y = 31104 / 10000

y = 3.1104 m above x axis.

extra 6m has to be added to get  the required length of wire

so required length of wire = 6 +3.1104 = 9.1104 m

or 9.11 m approximately.

 

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

8m wide means

 choose length of major axis  2a = 8m 

 or a =8/2 = 4m

semi ellipse , 2 m high at the centre

means length of semi minor axis , b =2

equation of ellipse is

[ (x^2) / (a^2) ] + [ (y^2) / (b^2)] = 1

[ (x^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

change given distance 1.5m to that in terms of centre (0,0) 

since a = 4

4-1.5 = 2.5 m

(2.5,y) lies on

[ (x^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

[ ((2.5)^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

  [ (y^2) / (2^2)] = 1 -[ ((2.5)^2) / (4^2) ]

  [ (y^2) / (2^2)] = 9.75/16

y^2 = 4*(9.75/16)

y^2 = (9.75/4)

y = [sqrt(9.75)]  / 2

y = 3.122/4

y =1.56 m

 

 

 

=================================================

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution

 

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

solution  

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means 

 

 

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

 

radius at the opening is 20/2 = 10

Choose the parabola as y^2 = 4ax

(5,10) lies on the parabola so that

10^2 = 4a(5)

or a = 100/20

a = 5

Therefore focus and centre are at  of 5 cm apart

In other words the focus is the midpoint of the given diameter.

 

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Choosing the parabola as (x^2)  = -4ay open downwards

because base is 5m wide at the base, (5/2) =  2.5m to each side  at 10 m depth

we can assume that (2.5,-10) lies on the parabola  (x^2)  = -4ay

so that (2.5) ^2 = -4 * a *(-10)

6.25 = 40 a

a =6.25 /40

resubstitute in (x^2)  = -4(6.25 /40)y

to get  (x^2)  = ( - 0.625 ) y

(x^2)  = ( - 0.625 )y

at 2 m distant, let (x,-2)  lie on (x^2)  = -40y

 (x^2)  = ( - 0.625 )(-2)

(x^2)  = 1.25

 (x^2)  = 125 / 100

 (x^2)  = 5 / 4

x =(sqrt(5)) / 2

Required width  = 2 * [(sqrt(5)) / 2] = sqrt(5)

Required width =2.23 m approximately

 

 

=================================================

ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution 

 

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

17.The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?

solution

13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.  

solution 

 

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution

 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

 

The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?

ncert cbse chapter 10  straight lines exercise 10.2 

 17.The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?

choose x as selling price per litre in rupees and y as demand in litres 

the line passes through (14,980 ) and (16,1220)

using 2 point form equation of the line is

[(y-980)/(1220-980)] = [(x-14)/(16-14)]

(y-980)/240 = (x-14)/2

y-980 = 120(x-14)

y-980 = 120x -1680 

 

y = 120x-1680+980

y = 120x -700

 

now put x = 17

y = 120*17 -700 

y = 1340 litres

 

13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9. 

let the intercepts be a and b

given a+b = 9

b = (9-a)

using intercept form of straight line

required equation is

[x/a] + [y/b] =1

change  b = (9-a) to get

[x/a] + [y/ (9-a)] =1--------------(1)

this passes through (2,2) 

so 

[2/a] + [2/ (9-a)] =1

 

[ 2(9-a)  +2a ] / [a(9-a)] = 1

[18 ]/ [a(9-a)] = 1

18 = [a(9-a)]

18 = 9a -(a^2) 

(a^2)  - 9a +18 = 0

(a-3)(a-6) =0

a = 3  or a=6

use in eqn(1) to get

2x+y =6  or x+2y =6

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

17.The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?

solution

13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.  

solution 

 

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution

 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find equation of the line.

 ncert cbse chapter 10  straight lines exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

let the intercepts of the line be a and b so that the line cuts the x axis and y axis at

(a,0)  and (0,b) respectively.

using section formula with ratio 1:2 ,

 ( ( 0 + 2a )  / (1+2) ,  (1b+0) / (2+1) ) = (h,k) by given condition

 so

2a/3 = h  and b/3 = k

a =3h/2  ; b= 3k

use equation of intercept form

(x/a) + (y/b) = 1

[x / (3h/2)] + [y / (3k)] = 1

2kx + hy = 3hk

 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

slope of the line passing through (2, 5) and (–3, 6)

is m1 = [6-5]  / [(-3) -2] = 1 /(-5)

 

condition for perpendicular lines is

m1 * m2 = (-1)

[1 /(-5)] * m2 = (-1)

so 

m2 = 5

is the slope of the required line.

required line passes through (-3,5) with slope 5

use point slope form to get the

equation of line as

y-5 =5[x+3] 

y-5 = 5x+15

5x-y+20 = 0

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution

 

 

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Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 ncert cbse chapter 10  straight lines exercise 10.2

 12. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 

Let the equation of the line be {x/a}  +{y/b} = 1

given intercepts are equal

put b = a

equation changes to {x/a}  +{y/a} = 1

so 

x + y = a --------------(1) 

(1) passes through (2,3)

so 2 + 3 = a

or a = 5

substitute back in (1)

equation changes to x + y = 5

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line. 

slope of the line joining (1, 0) and (2, 3) is

m1 = [3-0]/[2-1] = 3

using the condition for perpendicular lines

m1 * m2 = (-1)

3 * m2 = (-1)

m2 = (-1) / 3 is the slope of the required line

using section formula  the point which divides the line joining  (1, 0) and (2, 3)  in the ratio 1: n is given by

(  [(n+2) / (n+1)] , [(0 + 3) / (n+1)] ) = (  [(n+2) / (n+1)] , [(3) / (n+1)] )

required line passes through (  [(n+2) / (n+1)] , [(3) / (n+1)] ) 

with slope m2 = (-1) / 3 

 

using point slope form

equation  is 

[y - [(3) / (n+1)] ] = [(-1) / 3 ] *[x - [(n+2) / (n+1)] ]

3[n+1]y - 9 = -(n+1)x +(n+2)

or 

(n+1)x + 3[n+1]y = n+11

=================================================

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

solution 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means