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 3 x + y + 2 = 0.

 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.

[sqrt(3)] x + y + 2 = 0

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

we require the RHS to be no negative.

so multiply by (-1)

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

a = { -[sqrt(3)] } , b = (-1)

sqrt [(a^2) +(b^2)] = sqrt[3+1]  =sqrt[4] = 2

so divide each term in (1) with 2

 [ - {sqrt(3)} /2 ]x +(-1/2)y =1

use x cos θ + y sin θ = p 

to get 

cos θ =[ - {sqrt(3)} /2 ]

 sin θ = (-1/2)

so  θ is in the third quadrant

θ = pi + ( pi /6 ) =[7 pi/6]

using RHS

p =1

 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.

slope of the given line is 

m = [k-3] / [ 4 –( k^ 2)  ]

(a)

When parallel to x axis

slope = 0

hence [k-3] / [ 4 –( k^ 2)  ] = 0

k-3 = 0

k = 3

(b) Parallel to the y-axis,

 

When parallel to y axis

slope = infinity

hence [k-3] / [ 4 –( k^ 2)  ] = infinity

hence  

[ 4 –( k^ 2)  ] = 0

( k^ 2) = 4

k = 2 or (-2)

 

(c) Passing through the origin.

when line passes through the origin (0,0) 

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

 0 - 0 + (k^2) –7k + 6 = 0

(k^2) –7k + 6 = 0

[k-6][k-1]  = 0

k =6 , k = 1

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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 9 sequences and series miscellaneous exercise

 

 32.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished

solution

 

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