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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.

Q : 22     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

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From the figure above we can say that
The slope of line AC (m)= \tan \theta
Therefore,
\tan \theta = \frac{3-0}{5-a} = \frac{3}{5-a} \ \ \ \ \ \ \ \ \ \ (i)
Similarly,
The slope of line AB (m') = \tan(180\degree-\theta)
Therefore,
\tan(180\degree-\theta) = \frac{2-0}{1-a}
-\tan\theta= \frac{2}{1-a}
\tan\theta= \frac{2}{a-1} \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)
Now, from equation (i)  and (ii) we will get
\frac{3}{5-a} = \frac{2}{a-1}
\Rightarrow 3(a-1)= 2(5-a)
\Rightarrow 3a-3= 10-2a
\Rightarrow 5a=13
\Rightarrow a=\frac{13}{5}
Therefore, the coordinates of A.  is  \left ( \frac{13}{5},0 \right )

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