Question

A ray of light parallel to $y-$axis is incident on a parabola $y=4x^2$ polished from inside as shown in the figure. Find the co-ordinates $x\text{and}y$ of the point at which the incident ray should fall such that the reflected ray becomes horizontal after reflection.

• A. $(x,y)=(\frac{1}{8},\frac{1}{16})$
• B. $(x,y)=(\frac{1}{4},\frac{1}{8})$
• C. $(x,y)=(\frac{1}{16},\frac{1}{64})$
• D. $(x,y)=(\frac{1}{2},\frac{1}{4})$

Answer 1

The correct option is A $(x,y)=(\frac{1}{8},\frac{1}{16})$

The given situation can be shown as below
From the geometry we can say that the slope of the parabola at the point $(x,y)$ should be $\tan {45}^{\circ }$
Given, $y=4x^2$
$\Rightarrow \text{slope},\frac{dy}{dx}=8x=\tan {45}^{\circ }=1$
$\Rightarrow x=\frac{1}{8}$, putting value of $x$ in the equation of parabola, we will get
$y=4\times (\frac{1}{8}{)}^2=\frac{1}{16}$
Thus, $(x,y)=(\frac{1}{8},\frac{1}{16})$