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

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

$\dfrac 1f=\dfrac 1v+\dfrac 1u\\\implies \dfrac 1f=\dfrac {v+u}{vu}\\\implies vu=f(v+u)$

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

$(f+d_2)(f+d_1)=f(f+d_2+f+d_1)\\\implies f^2+fd_1+fd_2+d_1d_2=2f^2+fd_1+fd_2\\\implies 2f^2-f^2=d_1d_2\\\implies f^2=d_1d_2\\\implies f=\sqrt{d_1d_2}$

is the required focal length of the concave mirror.