Question

Two plane mirrors are placed perpendicular to each other and a point object $P$ is placed equidistant from both the mirrors. Find the equation of the geometerical shape at which the images of the object will lie.

• A. $x^2+y^2=2$
• B. $y=x$
• C. $x+y=2$
• D. $x^2+y^2=8$

Answer 1

The correct option is D $x^2+y^2=8$

${90}^{\circ }$ is the angle at which both mirrors are inclined.
So,
$n=\frac{{360}^{\circ }}{\theta }=\frac{{360}^{\circ }}{{90}^{\circ }}=4$
As $n$ is even, hence, the number of images formed,
$=n-1=4-1=3$
By symmetry, all the three images and the object will lie on a circle whose centre is at the point of intersection $(i.eO)$ of the mirrors and radius $\left(r\right)$ equal to the distance of the object $P$ from the intersection point.
So, $r^2=2^2+2^2=8$
[slope is $m=\tan {45}^{\circ }=1$, so, $x=y$]


We know the equation of the circle whose centre lie at the origin is given by,
$x^2+y^2=r^2$, where $r$ is radius.
Thus, here, the equation of the circle is,
$x^2+y^2=8$