Question 220 of 295

A convex lens of focal length $20cm$ is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is $10cm$. A point object is placed on principal axis at a distance of $60cm$ from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is _____ $cm$.

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Answer: 10

For lens $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$
$\Rightarrow \dfrac{1}{v}-\dfrac{1}{\left(-60\right)}=\dfrac{1}{20}\Rightarrow \dfrac{1}{v}+\dfrac{1}{60}=\dfrac{1}{20}$
$v=30cm$
For the final image to be formed on the object itself, after refraction from lens the ray should meet the mirror perpendicularly and the image by lens should be on the centre of curvature of mirror.
$R=30-10=20cm$.
Now the focal length of mirror ${f}_{m}=\dfrac{R}{2}=10cm$