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JEE MainMathematicsStraight Lines

A point A(3, 1) is fixed, and a variable point B lies on the line x + y = 6 . If the line segment AB subtends an angle of 4 at the origin O(0,0) , then the square of the distance between the two possible positions of point B is :

Options

  1. A20
  2. B180
  3. C200
  4. D30

Correct answer

C. 200

Step-by-step solution

The slope of the line OA is m₁ = 1 - 0 3 - 0 = 1 3 . Let the slope of the line OB be m . Since the angle between OA and OB is 4 , we have: ( 4 ) = | 1 3 - m 1 + m 3 | 1 = | 1 - 3m 3 + m | This gives two cases: Case 1: 1 - 3m 3 + m = 1 1 - 3m = 3 + m 4m = -2 m = - 1 2 Case 2: 1 - 3m 3 + m = -1 1 - 3m = -3 - m 2m = 4 m = 2 The equation of the line OB can be either y = - 1 2 x x + 2y = 0 or y = 2x . Point B lies on the line x + y = 6 . We find the intersection points: For x + 2y = 0 , substituting x = -2y into x + y =

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