Question

An object O is placed in front of a small plane mirror $M_1$ and a large convex mirror $M_2$ of local length $f$. The distance between O and $M_1$ is $x$, and the distance between $M_1$ and $y$. The image of O formed by $M_1$ and $M_2$ coincide. The value of $f$ is:

• A. $\frac{x^2-y^2}{2y}$
• B. $\frac{x^2+y^2}{2y}$
• C. $x-y$
• D. $\frac{x^2-y^2}{x+y}$

Answer 1

The correct option is A $\frac{x^2-y^2}{2y}$

An object $O$ is placed in front of a small plane mirror$=M_1$

and a large convex mirror$=M_2$ of focal length$=f$

The distance between $O$ and $M_1$ is $=x$

$u=x+y$ and

$v=x-y$

Using mirror formula,

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\frac{1}{f}=\frac{1}{x+y}-\frac{1}{x-y}\frac{1}{f}=\frac{1}{x+y}-\frac{1}{x-y}=\frac{x-y-x-y}{(x+y)(x-y)}=\frac{2y}{x^2-y^2}f=\frac{x^2-y^2}{2y}$