Question

An object is placed in front of a spherical mirror of focal length $f$. If $x$ and x' respectively represent the distance of the object and the image from the focus, then

• A. $f=x+x^{\prime }$
• B. $f^2=x{x}^{\prime }$
• C. $f=\left|x-x^{\prime }\right|$
• D. $f=x\pm x^{\prime }$ depending upon whether image is real or virtual

Answer 1

Answer: (B)

Object distance, $u=f+x$
Image distance, $v=f+x^{\prime }$
According to the mirror formula, $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$
$\frac{1}{f+x}+\frac{1}{f+x^{\prime }}=\frac{1}{f}$
$\Rightarrow \frac{f+x^{\prime }+f+x}{(f+x)\left(f+x^{\prime }\right)}=\frac{1}{f}$
$\left(2f+x^{\prime }+x\right)f=f^2+fx+fx+x{x}^{\prime }$
$2f^2+f{x}^{\prime }+fx=f^2+f{x}^{\prime }+fx+x{x}^{\prime }$
$f^2=x{x}^{\prime }$