Question

In a concave mirror, an object is placed a distance $d_1$ from the focus and the real image is formed at a distance $d_2$ from the focus. Then the focal length of the mirror is

• A. $\sqrt{d_1{d}_2}$
• B. $d_1{d}_2$
• C. $\frac{d_1{d}_2}{2}$
• D. $\sqrt{d_1/d_2}$

Answer 1

The correct option is A $\sqrt{d_1{d}_2}$

Given: In a concave mirror, an object is placed a distance $d_1$ from the focus and the real image is formed at a distance $d_2$ from the focus.

To find the focal length of the mirror

Solution:

Let the focal length of the concave mirror be $f$

And as the image formed is real, the object distance and image distance have same sign convention.

According to the given criteria,

Object distance, $u=f+d_1$

Image distance, $v=f+d_2$

Now by applying mirror formula, we get

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}⟹\frac{1}{f}=\frac{v+u}{vu}⟹vu=f(v+u)$

Substituting the values of $v$ and $u$, we get

$(f+d_2)(f+d_1)=f(f+d_2+f+d_1)⟹f^2+f{d}_1+f{d}_2+d_1{d}_2=2f^2+f{d}_1+f{d}_2⟹2f^2-f^2=d_1{d}_2⟹f^2=d_1{d}_2⟹f=\sqrt{d_1{d}_2}$

is the required focal length of the concave mirror.