$(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.

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