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(C) Let the object be at $O$. The distance of the object from the convex mirror is $u = -20 \, cm$.The plane mirror is placed $15 \, cm$ from the obje

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$20$

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(C) Let the object be at $O$. The distance of the object from the convex mirror is $u = -20 \, cm$.The plane mirror is placed $15 \, cm$ from the object. Since the object is $20 \, cm$ from the convex mirror,the plane mirror is $20 - 15 = 5 \, cm$ behind the convex mirror.The image formed by the plane mirror is at a distance of $15 \, cm$ behind the plane mirror. Since the plane mirror is $5 \, cm$ behind the convex mirror,the image is at a distance of $15 + 5 = 20 \, cm$ behind the convex mirror.Thus,the image distance for the convex mirror is $v = +20 \, cm$.Using the mirror formula $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$:$\frac{1}{f} = \frac{1}{20} + \frac{1}{-20} = 0$. This implies $f = \infty$,which is incorrect for a convex mirror. Let us re-evaluate.Correction: The image formed by the plane mirror is $15 \, cm$ behind the plane mirror. The plane mirror is $5 \, cm$ behind the convex mirror. So the image is $15 - 5 = 10 \, cm$ behind the convex mirror. Thus $v = +10 \, cm$.Now,$\frac{1}{f} = \frac{1}{10} + \frac{1}{-20} = \frac{2-1}{20} = \frac{1}{20}$.Therefore,$f = 20 \, cm$.

An object is placed $20 \, cm$ in front of a convex mirror. $A$ plane mirror is also placed such that the virtual images formed by both mirrors coincide. If the plane mirror is placed $15 \, cm$ from the object,find the focal length of the convex mirror in $cm$.

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