Question

A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and it passes through the point (5, 3). Find the co-ordinates of the point A

Answer 1

Let P(1, 2) and (5, 3) are the given points.

By the property of reflection,

∠XAB = ∠OAP = θ

(Angle of incidence = Angle of reflection)

Slope of the line OA(x – axis) m₁ = 0

Slope of the line joining the points P(1, 2) and A(x, 0)

Slope of AP, m₂ = `(2 - 0)/(1 - x) = 2/(1 - x)`

Slope of the line joining the points B(5, 3 ) and A(x, 0)

tan θ = `|("m"_1 + "m"_3)/(1 + "m"_1 "m"_3)|`

= `|(0 - 3/(5 - x))/(1 + 0(3/(5 - x)))|`

tan θ = `|(3/(5 - x))/1|`

= `3/(5 - x)`  ......(1)

tan ∠XAB = `|("m"_1 + "m"_2)/(1 + "m"_1 "m"_2)|`

tan(180° – θ) = `|(0 - 2/(1 - x))/(1 - 0 xx 2/(1 - x))|`

– tan θ = `|(2/(1 - x))/1|`

= `2/(1 - x)`

tan θ = `- 2/(1 - x)`  ......(2)

From equations (1) and (2)

`3/(5 - x) = - 2/(1 - x)`

3(1 – x) = – 2(5 – x)

3 – 3x = – 10 + 2x

2x + 3x = 10 + 3

5x = 13

⇒ x = `13/5`

∴ The required point A is `(13/5, 0)`