Question

A ray parallel to $x$ - axis is incident at a point $P$ on a parabolic reflecting surface, and the reflected ray becomes parallel to $y$ - axis as shown in the figure. If the equation of parabola is given by $y^2=2x$, then the coordinates of point $P$ are

• A. $(1,\frac{1}{2})$
• B. $(\frac{1}{2},1)$
• C. $(\frac{3}{2},1)$
• D. $(\frac{1}{3},\frac{1}{2})$

Answer 1

The correct option is B $(\frac{1}{2},1)$

Given equation,
$y^2=2x$
Differentiate w.r.t $x$
$2y\frac{dy}{dx}=2$
$\frac{dy}{dx}=\frac{1}{y}$
$\frac{1}{y}={\left(\frac{dy}{dx}\right)}_P=\tan \theta =\tan {45}^{\circ }=1$
$y=1$
Substitute $y=1$ in $y^2=2x\Rightarrow x=\frac{1}{2}$