Question

A ray of light passing through the point B ( 1,2 ) reflects on the X-axis at point A and the reflects ray passes through the point C ( 5, 6 ) . Find the coordinate of the point A.

Answer 1

Let the coordinates of the point A be ( a,0) .
Let AL be the perpendicular to xx - axis.
By Law of reflection we know,
angle of incidence = angle of reflection.
Hence Let =180∘−[θ+2(90∘−θ)=180∘−[θ+2(90∘−θ)
(∵ϕ=90−θ)(∵ϕ=90−θ)
∴=θ⇒Slope of the line AC,m=3−05−aAC,m=3−05−a
also m=tanθm=tan⁡θ
⇒tanθ=35−a⇒tan⁡θ=35−a-------(1)
Slope of the line AB=2−01−aAB=2−01−a
∴tan(180∘−θ)=21−a∴tan⁡(180∘−θ)=21−a
But tan(180∘−θ)=−tanθtan⁡(180∘−θ)=−tan⁡θ
∴−tanθ=21−a∴−tan⁡θ=21−a
∴tanθ=2a−1∴tan⁡θ=2a−1------(2)
equally (1) and (2) we get,
35−a35−a=2a−1=2a−1
⇒3(a−1)=2(5−a)⇒3(a−1)=2(5−a)
3a−3=10−2a3a−3=10−2a
⇒5a=13⇒5a=13
∴a=135∴a=135
Hence the coordinate of A are (135(135,0)