Q85 / 270 · NDA 2023 (Phase 1) · General Ability
A rectangle $A B C D$ is kept in front of a concave mirror of focal length $f$ with its corners $A$ and $B$ being, respectively, at distances $2 f$ and $3 f$ from the mirror with $A B$ along the principal axis as shown in the figure. It forms an image $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ in front of the mirror. What is the ratio of $B^{\prime} C^{\prime}$ to $A^{\prime} D^{\prime \prime}$
- A$1$
- B$2$
- Correct answerC$\frac{1}{2}$
- D$\frac{2}{3}$
Solution
$A B C D$ is a rectangle, then $A D=B C----(1)$
Given that corners, $A$ and $B$ are at distances $2 f$ and $3 f$ from the mirror, respectively, along the principal axis, then,
$u$ for corner $A=2 f$
From mirror equation,
$\frac{1}{f}=\frac{1}{v}+\frac{1}{2 f}$
$\Rightarrow v=2 f$
Magnification $m=\frac{h^{\prime}}{h}=-\frac{v}{u}=-\frac{2 f}{2 f}=-1$
This means size of the image is equal to size of the object,
$\Rightarrow A D=A^{\prime} D^{\prime}----(2)$
Now if for corner $\mathrm{B}=3f$
From the mirror equation,
$\frac{1}{f}=\frac{1}{v}+\frac{1}{3 f}$
$\Rightarrow v=\frac{3}{2} f$
Magnification, $m=\frac{h^{\prime}}{h}=-\frac{v}{u}=-\frac{\frac{3}{2} f}{3 f}=-\frac{1}{2}$
Means size of the image is half to size of the object,
$\Rightarrow B^{\prime} C^{\prime}=\frac{1}{2} B C----(3)$
From equations 1,2 and 3 , the ratio of $B^{\prime} C^{\prime}$ to $A^{\prime} D^{\prime}$ is,
$\Rightarrow \frac{B^{\prime} C^{\prime}}{A^{\prime} D^{\prime}}=\frac{\frac{1}{2} B C}{A D}=\frac{1}{2}$
Hence, the correct answer is option: 'C'.