Question

An object A is placed on the principal axis of a convex mirror at a distance of $60cm$ in front of it. A plane mirror is inserted between the object and the convex mirror at a distance $40cm$from the object with the reflecting surface of the plane mirror facing the object. If the images formed by the two mirrors coincide, find the focal length of the convex mirror.

$\begin{align}
  & \text{A}\text{. }15cm \\
 & \text{B}\text{. }25cm \\
 & \text{C}\text{. }20cm \\
 & \text{D}\text{. }30cm \\
\end{align}$

Answer 1

Hint: In a plane mirror the image is always virtual, erect and of the same size that of the object and the image is situated at the same distance from the mirror as that of the object. Calculate the image distance of the plane mirror from that you can calculate the image distance from the convex mirror and use mirror formula to calculate the focal length of the convex mirror.

Formula used:
The mirror formula is given by \[\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}\].where,
$\begin{align}
  & u=\text{ object distance from the mirror} \\
 & v=\text{ image distance from the mirror} \\
 & f=\text{focal length of the mirror} \\
\end{align}$

Complete step by step answer:
Given that the object is situated on the principal axis of a convex mirror at a distance of $60cm$. So by taking sign convention
$u=-60cm$.
Now a plane mirror is inserted between the object and the convex mirror at a distance $40cm$from the object with the reflecting surface of the plane mirror facing the object. So the plane mirror is situated at a distance $60cm-40cm=20cm$from the convex mirror.

From the properties of plane mirrors we know that the image formed by a plane mirror is virtual, erect situated at the opposite side of the object and at the same distance as that of the object.
So the image of the plane mirror is situated at $40cm$ right to the plane mirror. The distance of this image from the convex mirror is $-20cm+40cm=+20cm$. i.e the image of the plane mirror is situated $20cm$right to that of the convex mirror.
According to the question the image due to the convex mirror coincides with the image with the plane mirror. So the image of the object A by the convex mirror is situated $20cm$right to that of convex mirror. i.e. $v=+20cm$
So we got $u=-60cm$and $v=+20cm$
Applying mirror formula
\[\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}\]
Taking sign convention and putting the values in the above mirror equation we got.
$\begin{align}
 & \dfrac{1}{20}+\dfrac{1}{-60}=\dfrac{1}{f} \\
 & \Rightarrow \dfrac{1}{f}=\dfrac{1}{20}-\dfrac{1}{60}=\dfrac{3-1}{60}=\dfrac{2}{60} \\
 & \Rightarrow f=\dfrac{60}{2}=30cm \\
\end{align}$

So the focal length of the convex mirror is found to be $30cm$ situated right to that of the convex mirror. So the correct option is D.

Note:
Note that for a convex mirror the value of focal length will always be positive by sign convention. By sign convention, if we move right from the mirror then that distance will be taken as positive and if we move left from the mirror the distance will be taken as negative. Similarly moving vertically upward positive and moving downward negative.