Three plane mirrors form an equilateral triangle with each side of length 𝐿. There is a small hole at a distance 𝑙 > 0 from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle 𝜃 and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane. Which of the following statement(s) is(are) correct? (A) The ray of light will come out for 𝜃 = 30∘, for 0 < 𝑙 < 𝐿. (B) There is an angle for 𝑙 = � � at which the ray of light will come out after two reflections. (C) The ray of light will NEVER come out for 𝜃 = 60∘ , and 𝑙 = � � . (D) The ray of light will come out for 𝜃 = 60∘ , and 0 < 𝑙 < � � after six reflections.

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 We analyze the problem step by step to determine which statements are correct.

Step 1: Understanding the setup

  • Three plane mirrors form an equilateral triangle with side length L.

  • A small hole is located at a distance l>0 from one of the corners.

  • A ray of light enters through the hole at an angle θ and can only exit through the same hole.

  • The cross-section of the mirror configuration and the ray of light lie on the same plane.

Step 2: Analyzing the path of the ray

For the ray to exit through the same hole, it must undergo reflections that bring it back to the hole. The number of reflections depends on the angle θ and the distance l.

Step 3: Checking each statement

(A) The ray of light will come out for θ=30∘, for 0<l<L.

  • For θ=30∘, the ray will reflect symmetrically in the equilateral triangle. After six reflections, the ray will return to the hole.

  • Statement (A) is correct.

(B) There is an angle for l=L2 at which the ray of light will come out after two reflections.

  • For l=L2, the ray can be directed at an angle such that it reflects twice and returns to the hole.

  • Statement (B) is correct.

(C) The ray of light will NEVER come out for θ=60∘, and l=L2.

  • For θ=60∘, the ray will reflect in such a way that it does not return to the hole after any number of reflections.

  • Statement (C) is correct.

(D) The ray of light will come out for θ=60∘, and 0<l<L2 after six reflections.

  • For θ=60∘, the ray will not return to the hole after six reflections.

  • Statement (D) is incorrect.

Final Answer:

Statements (A), (B), and (C) are correct.

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