Question

Total distance travelled by an incident ray and the corresponding reflected ray is the least if the point of incidence coincides with

• A. an end of the minor axis
• B. an end of the major axis
• C. an end of the latus rectum
• D. None of these

Answer 1

The correct option is D None of these

Light ray from $\lambda (x+2)-y+2=0-\left(1\right)$ reflects from elliptical surface and passes through $\mu (x-2)+(-(y-2))=0-\left(2\right)$

Equation $1$ is family of lines passing through $(-2,2)$

Equation $2$ is family of lines passing through $(2,2)$

Light ray from one focus will pass through another focus. $\therefore (-2,2),(2,2)$ are focus of ellipse with $y=2$ as major axis and corner at $(0,2)$. And also, foot of perpendicular from $(2,2)$ focus upon any tangent to the ellipse lies on $x^2+{(y-2)}^2=9\Rightarrow$ auxiliary circle.

$ae=2,a=3$

$a^2-b^2=4,a=3$

$b=\sqrt{5}$

$\Rightarrow \frac{x^2}{9}+\frac{{(y-2)}^2}{5}=1$

$P(3\cos \theta ,2+5\sin \theta )$ is any point on ellipse light from focus $S^{\prime }(-2,2)$ touches $P$ and gets reflected and passes through $(2,2)$

$\Rightarrow S^{\prime }P+SP=2a$ is always contact irrespective of the location $P$.