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JEE MainMathematicsStraight Lines

A ray of light travelling along the line 2x + y = 5 is reflected by a plane mirror L . If the reflected ray travels along the line x - 3y = 2 , then the sum of all possible distinct values for the slope of the mirror L is

Options

  1. A-2
  2. B- 5 3
  3. C-1
  4. D2

Correct answer

D. 2

Step-by-step solution

Let the slope of the mirror L be m . By the laws of reflection, the mirror makes equal angles with the incident ray and the reflected ray. Thus, it acts as an angle bisector for the lines L₁: 2x + y = 5 and L₂: x - 3y = 2 . The slope of L₁ is m₁ = -2 and the slope of L₂ is m₂ = 1 3 . Since the angle made by the mirror with both lines is the same, we have: | m - m₁ 1 + m m₁ | = | m - m₂ 1 + m m₂ | | m + 2 1 - 2m | = | m - 1/3 1 + m/3 | = | 3m - 1 3 + m | Opening the modulus with a positive sign yields: m + 2 1 - 2m

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