Question

A ray of light passing through the point (1, 2) is reflected on the x-axis at a point p and passes through the point (5, 3). The abscissa of the point p is:

Answer 1

heyaa!!
here is your answer..
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 <BAL=<CAL=ϕ<BAL=<CAL=ϕ

Let <CAX=θ<CAX=θ

<OAB=180∘−(θ+2ϕ)<OAB=180∘−(θ+2ϕ)

=180∘−[θ+2(90∘−θ)=180∘−[θ+2(90∘−θ)

(∵ϕ=90−θ)(∵ϕ=90−θ)

∴<OAB=180∘−θ−180∘+2θ∴<OAB=180∘−θ−180∘+2θ

=θ⇒<BAX=180∘−θ=θ⇒<BAX=180∘−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)
instead of A u can write P...
hope it helps..