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(C) For the convex mirror,the object distance $u = -30 \ cm$ and focal length $f = +30 \ cm$.Using the mirror formula $\frac{1}{v} + \frac{1}{u} = \fr

Vedclass · Accepted Answer

$22.5$

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(C) For the convex mirror,the object distance $u = -30 \ cm$ and focal length $f = +30 \ cm$.Using the mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$:$\frac{1}{v} + \frac{1}{-30} = \frac{1}{30}$$\frac{1}{v} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}$So,$v = +15 \ cm$. This image is formed $15 \ cm$ behind the convex mirror.The distance of this image from the object is $30 \ cm + 15 \ cm = 45 \ cm$.For the plane mirror to produce an image that coincides with the image from the convex mirror,the plane mirror must be placed such that the object distance from the plane mirror equals the image distance from the plane mirror.Let the plane mirror be at a distance $x$ from the object. The image formed by the plane mirror is at a distance $x$ behind the plane mirror.The distance of this image from the convex mirror is $(30 - x) + x = 30 \ cm$ (relative to the plane mirror position).For the images to coincide,the plane mirror must be placed such that its image is at the same position as the convex mirror's image ($15 \ cm$ behind the convex mirror).Let $d$ be the distance between the mirrors. The object is $30 \ cm$ from the convex mirror. Let the plane mirror be at distance $d$ from the convex mirror. The object is at distance $(30 - d)$ from the plane mirror.The image formed by the plane mirror is at distance $(30 - d)$ behind the plane mirror,which is at a distance $d + (30 - d) = 30 \ cm$ from the convex mirror.Wait,the condition is that the images coincide. The convex mirror forms an image at $15 \ cm$ behind its pole. The plane mirror must form an image at the same location.If the plane mirror is at distance $d$ from the convex mirror,the object is at distance $(30 - d)$ from the plane mirror. The image is at distance $(30 - d)$ behind the plane mirror.Distance of this image from the convex mirror pole $= d + (30 - d) = 30 \ cm$ (in front of the convex mirror).This is not correct. Let the plane mirror be at distance $d$ from the convex mirror. The object is at distance $(30-d)$ from the plane mirror. The image is at distance $(30-d)$ behind the plane mirror. The distance of this image from the convex mirror is $d + (30-d) = 30 \ cm$ behind the plane mirror,which is $30-d$ from the convex mirror.Equating the image positions: $15 = -(30 - 2d) \Rightarrow 15 = -30 + 2d \Rightarrow 2d = 45 \Rightarrow d = 22.5 \ cm$.

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