Question

A parabolic mirror is kept along $y^2=4x.$ A ray along $y=3$ strikes the mirror (from the positive direction of $x-$axis) and gets reflected along the line $L=0.$ Then which of the following is NOT true ?

• A. The line $L=0$ passes through $(0,-\frac{12}{5}).$
• B. The line $L=0$ intersects the $x-$axis at $(1,0).$
• C. The line $L=0$ is a tangent to the circle $x^2+y^2=1.$
• D. The line $L=0$ makes acute angle with positive $x-$axis in anticlockwise direction.

Answer 1

The correct option is C The line $L=0$ is a tangent to the circle $x^2+y^2=1.$


Equation of mirror $:y^2=4x$
$y=3$ meets the parabola at $P(\frac{9}{4},3)$
We know, the reflected ray passes through the focus $(1,0).$
Equation of $PS:y=\frac{12}{5}(x-1)$
$\Rightarrow L\equiv 12x-5y-12=0$

$L=0$ passes through $(0,-\frac{12}{5}).$

$L=0$ intersects the $x-$axis at $(1,0).$

Given circle $:x^2+y^2=1,$
Centre $\equiv (0,0)$ and radius $=1$
Distance of centre $(0,0)$ from the line $L=0$
$=\left|\frac{-12}{\sqrt{{12}^2+(-5{)}^2}}\right|=\frac{12}{13}\ne$ radius
$\therefore L=0$ is not a tangent to the given circle.

Slope of $L=\frac{12}{5}>0$
$\therefore L=0$ makes an acute angle with positive $x-$axis in anticlockwise direction.