Question

Two rays of light coming along the lines $y=1$ and $y=-2$ from the positive direction of x-axis and strike a concave mirror whose intersection with the xy-plane is a parabola $y^2=x$ at $A$ and $B$ respectively. The reflected rays pass through a fixed point $C$, then the area of the $△ABC$ is

• A. $\frac{21}{8}$ sq. unit
• B. $\frac{19}{2}$ sq. unit
• C. $\frac{17}{2}$ sq. unit
• D. $\frac{15}{2}$ sq. unit

Answer 1

The correct option is A $\frac{21}{8}$ sq. unit

The reflected rays will pass through the focus of the parabola as per definition.
Thus, point $C=(\frac{1}{4},0)$
Also, $A=(1,1)$ and $B=(4,-2)$
Thus, area of $\Delta ABC$ is $\frac{1}{2}\times$ $\left[\begin{array}{cccc}1& 4& \frac{1}{4}& 1\\ 1& -2& 0& 1\end{array}\right]$
$\Rightarrow$ Area $=\frac{1}{2}|-2+0+\frac{1}{4}-4+\frac{1}{2}-0|=\frac{1}{2}\left|\frac{-21}{4}\right|=\frac{21}{8}$