A ray of light passing through the point A and the reflected ray passes 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.

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Solution

Let BA be the incident ray and AC be the reflected ray.
Now for line AC

$t a n ϕ = \frac{3 - 0}{5 - x}$

$\Rightarrow t a n ϕ = \frac{3}{5 - x} \dots (i)$

Now for line BA
$t a n (180 - ϕ) = \frac{2 - 0}{1 - x}$

$\Rightarrow - t a n ϕ = \frac{2}{1 - x} \dots (i i)$

From (i) and (ii) we have

$\frac{3}{5 - x} = \frac{- 2}{1 - x}$

$\Rightarrow 3 - 3 x = - 10 + 2 x$

$\Rightarrow - 5 x = - 13 \Rightarrow x = \frac{13}{5}$

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

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