Question

If a ray of light passing through $(2,2)$ reflects on the $x-$axis at a point $P$ and the reflected ray passes through the point $(6,5)$, then the co-ordinates $P$ is

• A. $(\frac{13}{7},0)$
• B. $(\frac{22}{7},0)$
• C. $(\frac{25}{7},0)$
• D. $(\frac{27}{7},0)$

Answer 1

The correct option is B $(\frac{22}{7},0)$

Let $P(h,0)$ and $PN$ be normal at $P$
$\angle APN=\angle NPB=\theta$


So, we get
Slope of $PB=\tan ({90}^{\circ }-\theta )$
$\Rightarrow \frac{5-0}{6-h}=\cot \theta \Rightarrow \frac{5}{6-h}=\cot \theta \cdots \left(1\right)$
And
Slope of $AP=\tan (90+\theta )$
$\Rightarrow \frac{2-0}{2-h}=-\cot \theta \Rightarrow \frac{2}{2-h}=-\cot \theta$
Using equation $\left(1\right)$, we get
$\Rightarrow \frac{2}{2-h}=-\left(\frac{5}{6-h}\right)\Rightarrow 12-2h=-10+5h\Rightarrow h=\frac{22}{7}$

Hence, the coordinates of $P$ is $(\frac{22}{7},0)$