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

Let the coordinates of point A be (a, 0).
Draw a line (AL) perpendicular to the x-axis.
We know that angle of incidence is equal to angle of reflection.
Hence, let $∠BAL = ∠CAL = \phi$

Let $∠CAX = θ.$

$∴∠OAB = 180° \- (θ + 2\phi) = 180° \- [θ + 2(90°- θ)]$
$= 180° \- θ \- 180° + 2θ = θ$
$∴∠BAX = 180° \- θ$

Now, slope of line $AC=\frac{3-0}{5-a}$

$⇒ tanθ=\frac{3}{5-a} ........(1)$

Slope of line $AB =\frac{2-0}{1-a}$

$⇒ tan(180°-θ)=\frac{2}{1-a}$

$⇒ -tanθ=\frac{2}{a-1} ...(2)$
From equations (1) and (2), we obtain
$\frac{3}{5-a}=\frac{2}{a-1}$

$⇒ 3a – 3 = 10 – 2a$

$⇒ a = \frac{13}{5}$

Thus, the coordinates of point A are $(\frac{13}{5}, 0).$