Question

The light rays emanating from a point source situated at origin when reflected from the mirror of a search light are reflected as beam parallel to the x-axis. The surface is .

• A. circular
• B. elliptical
• C. parabolic
• D. hyperbolic

Answer 1

The correct option is C parabolic

Equation of normal at P
$Y-y=-\frac{1}{m}(X-x)$
Substitute $Y=0\Rightarrow X=x+my$
Now
${\left(OP\right)}^2={\left(ON\right)}^2{x}^2+y^2={(x+my)}^2\Rightarrow m=-\frac{x+\sqrt{x^2-y^2}}{y}=\frac{dy}{dx}$
Substitute $y=vx$
$\int \frac{vdv}{\sqrt{1+v^2}(\pm 1-\sqrt{1-v^2})}=\int \frac{dx}{x}$
Substitute $\pm 1-\sqrt{1-v^2}=t$
$-\int \frac{dt}{t}=\ln x+c$
Hence $y^2=c^2\pm 2cx$