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

Answer 1

Answer: (B)

Object distance, $u=f+x$
Image distance, $v=f+x'$
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'+f+x}{(f+x)\left(f+x'\right)}=\frac{1}{f} $
$\left(2 f+x'+x\right) f=f^{2}+f x +f x+x x' $
$2 f^{2}+f x'+f x=f^{2}+f x'+f x+xx'$
$f^{2}=x x'$