Question

An object is gradually moving away from the focal point of a concave mirror along the axis of the mirror. The graphical representation of the magnitude of linear magnification $\left(m\right)$versus distance of the object from the mirror $\left(x\right)$ is correctly given by (Graphs are drawn schematically and are not to scale)

• A.
• B.
• C.
• D.

  

Answer 1

The correct option is B

Step 1. Given data:

Magnitude of linear magnification, $m$

Distance of the object from the mirror, $u=-x$

Let $f$ is focal length, $v$ is the distance of the image from the mirror.

Let $f=-F_0$ the focal length will be negative for the concave mirror,

Step 2. Using mirror formula:

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

$f=-F_{0,}v=?,u=-x$

$\Rightarrow$ $-\frac{1}{F_0}=\frac{1}{v}+\left(\frac{1}{-x}\right)$

$\Rightarrow$$-\frac{1}{F_0}+\frac{1}{x}=\frac{1}{v}$

$\Rightarrow$ $v=-\left(\frac{x{F}_0}{x-F_0}\right)$

Step 3. Substitute $V$ value in the formula of the magnitude of linear magnification:

$m=-\frac{\left(v\right)}{\left(u\right)}$

$\Rightarrow$$m=\frac{\left(-x{F}_{\circ }\right)}{(x-F_{\circ })(-x)}$

$\Rightarrow$$m=\frac{F_{\circ }}{x-F_{\circ }}$

$\Rightarrow$$m\propto \frac{1}{x}$

Hence, the curve is a rectangular hyperbola.

$Ifx=F_{\circ },m\to \infty$

$Ifx=2F_{\circ },m=1$

Hence, the correct option is (B).