1. Image formed by the convex mirror

For a convex mirror, using the mirror formula

with Cartesian sign convention:

• object distance: $u=-30\text{ cm}$
• focal length of convex mirror: $f=+30\text{ cm}$

So,

Thus, the convex mirror forms a virtual image $15$ cm behind it.

2. Condition for coincidence with image formed by plane mirror

Let the distance between the convex mirror and plane mirror be $d$.

The object is $30$ cm in front of the convex mirror. Since the plane mirror is placed on the other side, the distance of the object from the plane mirror is

(assuming the plane mirror lies between the object and convex mirror, which is the only physically meaningful arrangement for coincidence here).

A plane mirror forms image at the same distance behind it as the object is in front of it. Therefore, the plane mirror image is at a distance

behind the plane mirror.

Hence, measured from the convex mirror, the position of this image is

which is independent of $d$.

This does not match the convex mirror image at $15$ cm, so this arrangement is impossible.

3. Correct geometry

So the plane mirror must be placed behind the object, on the same side as the object, at distance $d$ from the convex mirror.

Then the object is between the plane mirror and convex mirror.

Distance of object from plane mirror:

Since the object is in front of the convex mirror by $30$ cm and plane mirror is at distance $d$ from the convex mirror on the object side.

The image in the plane mirror is formed behind the plane mirror by the same amount, so its distance from the convex mirror is

Again not useful.

So instead, let us interpret coincidence as follows: the image formed by the plane mirror acts as object for convex mirror, and final image coincides with the image that convex mirror alone would form.

If the plane mirror is at distance $d$ in front of the convex mirror, then object distance from plane mirror is

So image in plane mirror is at equal distance behind it, i.e. at distance from convex mirror

from the convex mirror on the object side if $2d<30$.

Thus this image acts as an object for the convex mirror at distance

from the convex mirror.

For the final image in the convex mirror to coincide with the previous convex-mirror image position $v=15$ cm, we use mirror formula with same final image distance $v=15$:

So the effective object for the convex mirror must be at $30$ cm in front of it.

That means the plane mirror must produce the image at the original object position, which is not possible unless special overlap occurs.

4. Standard shortcut for this configuration

The convex mirror forms image $15$ cm behind it. For the plane mirror image to coincide with this, the plane mirror must be placed such that the image due to plane mirror lies $15$ cm behind the convex mirror.

If the plane mirror is at distance $d$ in front of the convex mirror, object distance from plane mirror is $30-d$. Hence plane mirror image is $(30-d)$ behind the plane mirror, i.e. at position from convex mirror:

behind the convex mirror.

For coincidence,

So the distance between the mirrors is

5. Check options

• A: $45$ cm — incorrect
• B: $15$ cm — incorrect
• C: $22.5$ cm — correct
• D: $7.5$ cm — incorrect

Therefore, the correct answer is