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.

 

 ncert cbse chapter 10 straight lines miscellaneous exercise

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.

Since A lies on the x axis, the y coordinate of A is 0

Let A =(k,0) 

let P=(1,2)

Q =(5,3)

both PA and AQ make equal angles with the normal line (perpendicular to the x axis) at A due to  the law of reflection.

If AQ makes an angle u with the positive direction of the x axis, then PA makes an angle of (180-u) with the positive direction of the x axis.

using the definition of slope

 

for PA, P=(1,2), A =(k,0)

tan(u) =[0-2] / [k-1] --------------------(1)

for QA , Q =(5,3) , A =(k,0)

tan[180-u]  = [0-3] / [k-5]

but tan[180-u]  = {-tan (u)}

we get 

{-tan (u)} = [0-3] / [k-5] ------------(2)

using eqn (1)  in  (2)

-{[0-2] / [k-1] } = [0-3] / [k-5]

2(k-5) = (-3)(k-1)

2k-10 = -3k+3

5k =13

k =[13/5]

therefore

A=( (13/5)  , 0 )

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

 

first line is

[ 9x + 6y – 7 = 0 ] dividing  by 3 to make the coefficients in the two given equations the same

3x +2y -(7/3) =0 -------------(1)

second line is  

 3x + 2y + 6 =0 --------------(2)

 

A line parallet to these lines will be of the form

3x+2y +k =0 ----------------(3) 

distance between (1) and (3)

d1 = | k +(7/3) | / sqrt[(3^2) +(2^2)]

distance between (2) and (3)

d2 = | k -6 | / sqrt[(3^2) +(2^2)]

 

for equidistant line d1 = d2

 equating d1 and d2 and cancelling off the denominator

| k +(7/3) | = | k -6 | 

k +(7/3) =  k -6   OR  k +(7/3) = -(k -6 )

(7/3) =(-6) which is not possible

OR 

k +(7/3) = -(k -6 )

2k =6 -(7/3) 

2k = (11/3)

k = (11/6)

substitute in (3)

3x+2y +(11/6) =0

 

18 x  + 12y +11 =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

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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