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A thin rod is placed along the principal axis of a concave mirror of focal length 20 cm . The rod is positioned such that its real image spans the exact same segment of the principal axis as the rod itself. If the midpoint of the rod is at a distance of 45 cm from the pole of the mirror, the length of the rod is ______ cm .

Correct answer

30

Step-by-step solution

Let the coordinates of the two ends of the rod be u_A and u_B . For the image to span the exact same segment as the object, the image of end A must form at end B, and the image of end B must form at end A. This implies that u_A and u_B are conjugate positions. Using the mirror formula: 1 u_A + 1 u_B = 1 f The focal length of the concave mirror is f = -20 cm . The midpoint of the rod is at 45 cm from the pole, so: u_A + u_B 2 = -45 u_A + u_B = -90 cm Substituting this into the mirror formula rearranged as u_A + u_B

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