Question

a plane mirror is placed at a distance 12.5 cm from the focus of a concave mirror of radius curvature 20cm find where the object can be placed between the two Mirrors of the first image in the both of the Mirrors concide

Answer 1

Radius of curvature of the mirror is $(R)=20cm$

Hence, focal length of concave mirror, $f =\frac{-R}{2}= -10cm$

As per the question, plane mirror is placed at a distance of 12.5 cm from the focus of concave mirror.

Let an object is placed x distance from plane mirror.

then, image distance from plane mirror = x cm right side of plane mirror.

so, image distance $v= -(12.5 + 10 - x) =-(22.5-x)$ from concave mirror

again,

object distance from concave mirror, $u = -(x + 12.5 + 10) = -(x+ 22.5)cm$

using mirror formula,

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

$\Rightarrow \frac{-1}{(22.5-x)} -\frac{ 1}{(x+22.5)} = \frac{1}{-10}$

$\Rightarrow \frac{-(22.5+x)-(22.5-x))}{(22.5^2-x^2)} = \frac{1}{-10}$

$\Rightarrow \frac{-45}{(22.5^2-x^2)} = \frac{1}{-10}$

$\Rightarrow 450=22.5^2-x^2$

$\Rightarrow x^2=506.25-450$

$\Rightarrow x^2=56.25cm^2$

$\Rightarrow x=\sqrt{56.25}cm$

$\Rightarrow x=7.5cm$

hence, object is placed 7.5 cm left side from plane mirror.