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 \((\frac{13}{5}, 0).\)