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Difficulty: Very hardReflection of Light at Plane and Curved Mirrors

An object is placed at a distance uu in front of a concave mirror of focal length 12 cm12\text{ cm}. A plane mirror is placed perpendicular to the principal axis at a distance of 32 cm32\text{ cm} in front of the concave mirror, between the object and the concave mirror. If the real image formed by the concave mirror coincides in space with the virtual image formed by the plane mirror, what is the value of uu in centimeters?

Answer: 48 cm

Answer

The correct object distance uu is 48 cm48\text{ cm}.
The object is located at distance uu from the concave mirror. With the plane mirror at 32 cm32\text{ cm} from the concave mirror, the object distance from the plane mirror is u−32u - 32. The plane mirror forms an image at distance u−32u - 32 behind itself, which corresponds to 32−(u−32)=64−u32 - (u - 32) = 64 - u from the concave mirror. Setting v=64−uv = 64 - u in the mirror formula 112=1u+164−u\frac{1}{12} = \frac{1}{u} + \frac{1}{64 - u} gives u2−64u+768=0u^2 - 64u + 768 = 0. Factoring yields u=48 cmu = 48\text{ cm} or u=16 cmu = 16\text{ cm}. Because the plane mirror is between the object and the concave mirror, u>32 cmu > 32\text{ cm}, so u=48 cmu = 48\text{ cm}.

Step-by-Step Solution

1
Find the position of the image formed by the plane mirror in terms of uu.
The object is at a distance (u−32) cm(u - 32)\text{ cm} in front of the plane mirror. Its virtual image is formed (u−32) cm(u - 32)\text{ cm} behind the plane mirror, which places it at 32−(u−32)=(64−u) cm32 - (u - 32) = (64 - u)\text{ cm} in front of the concave mirror.
A plane mirror forms an image behind it at a distance equal to the object distance in front of it.
2
Equate the image distance of the concave mirror vv to the position of the plane mirror image.
v=64−uv = 64 - u
The question states that the image formed by the concave mirror coincides in position with the image formed by the plane mirror.
3
Substitute f=12 cmf = 12\text{ cm} and v=64−uv = 64 - u into the mirror equation.
112=1u+164−u\frac{1}{12} = \frac{1}{u} + \frac{1}{64 - u}
The standard mirror formula relates focal length, object distance, and image distance.
4
Solve the algebraic equation for uu.
112=(64−u)+uu(64−u)  ⟹  64u−u2=768  ⟹  u2−64u+768=0\frac{1}{12} = \frac{(64 - u) + u}{u(64 - u)} \implies 64u - u^2 = 768 \implies u^2 - 64u + 768 = 0
Combining fractions and multiplying across gives a quadratic equation in standard form.
5
Factor the quadratic equation and select the physical root.
(u−48)(u−16)=0  ⟹  u=48 cm(u - 48)(u - 16) = 0 \implies u = 48\text{ cm} or u=16 cmu = 16\text{ cm}. Since u>32 cmu > 32\text{ cm}, u=48 cmu = 48\text{ cm}.
The plane mirror is situated between the object and the concave mirror at 32 cm32\text{ cm}, so the object distance uu must be greater than 32 cm32\text{ cm}.

Key Concept

Image coincidence in combined plane and curved optical systems
Estimated Time:3m 0s
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Topics

Reflection of Light at Plane and Curved Mirrors

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