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An object is placed at an of 45 cm from a converging lens of focal length 30 cm . A mirror of radius of curvature 40 cm is to be placed on other side of lens so that the object coincides with its image. Find the distance of mirror from lens if mirror is convex.

$(a)\;20\;cm \\ (b)\;30\;cm \\ (c)\;50\;cm \\ (d)\;40\;cm$

the object and image will coincide only if light ray retraces its path and it will occur only when the ray normally strikes at mirror.
In other words , the centre of curvature of mirror and rays incident on mirror are collinear.
The rays after refraction from lens must be directed towards the centre of curvature of mirror at C.
If x is separation , then for lens,
$u= -45\;cm,v= x+40,f= 30\;cm$
Using lens formula,
$\large\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
or $\large\frac{1}{x+40}-\frac{1}{-45}=\frac{1}{30}$
or $x=\large\frac{45(30)}{45-30}$$-40 =50 \;cm$
Hence c is the correct answer.