Question

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.

Answer 1

The ray of light is passing through the point ( 1,2 ). The ray gets deflected from point A on the x axis and passes through the point ( 5,3 ).

Let the coordinate of the point A be ( h,0 ).

As the ray gets deflected from point A,

∠BAL=∠CAL=ϕ.

Let, ∠CAX=θ.

∠OAB=180°−( θ+2ϕ ) =180°−( θ+2( 90°−θ ) ) =180°−θ−180°+2θ =θ

Therefore, ∠BAX=180°−θ.

The formula for the slope of a line passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Substitute the values for ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( h,0 ) and ( 3,5 ) respectively in equation (1) to obtain the slope of AC.

tanθ= 3−0 5−h tanθ= 3 5−h (2)

Similarly, the slope of line AB is obtained by substituting the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( h,0 ) and ( 1,2 ).

tanθ= 2−0 1−h tan( 180°−θ )= 2 1−h −tanθ= 2 1−h tanθ= 2 h−1 (3)

By comparing equation (2) and (3), we get

3 5−h = 2 h−1 3( h−1 )=2( 5−h ) 3h−3=10−2h 3h+2h=10+3

Solve further,

5h=13 h= 13 5

Thus, the coordinates of point A are ( 13 5 ,0 ).