Question

A paraxial 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 the parabola is given by $y^2=2x$, then the coordinates of $P$ are

Answer 1

Since the reflected ray becomes perpendicular to the incident ray, we can conclude that the rays make angle ${45}^{\circ }$ each with normal at the point $P$.


Equation of parabola is given as $y^2=2x$
Differentiating both side w.r.t $x$,
$2y\frac{dy}{dx}=2$
$\Rightarrow \frac{1}{y}={\left(\frac{dy}{dx}\right)}_p=\tan \theta =\tan 45{}^{\circ }=1$
$\Rightarrow y=1$

Put $y=1$ in $y^2=2x\Rightarrow x=\frac{1}{2}$


Comparing above equation with $y^2=4ax$, we get $a=\frac{1}{2}$
So, focus of given parabola will be at $(a,0)=(\frac{1}{2},0)$.

And we know that reflected ray always passes through its focus and in this case, reflected ray is parallel to $y-$ axis.
So, $P(x,y)=P(\frac{1}{2},y)$.

Since, point $P$ lies on the parabola, so it will satisfy the equation of parabola.
$\therefore y^2=2\times \frac{1}{2}=1$
$\Rightarrow y=-1\text{or}+1$
For this case, $y=1$
$\therefore P(x,y)=(\frac{1}{2},1)$.