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JEE MainMathematicsStraight Lines

Two lines L₁ and L₂ pass through the point A(1,4) and each makes an angle of 4 with the line 2x - y = 0 . These two lines intersect a third line L₃ : x + y = 2 at points B and C , forming a triangle ABC . If (h, k) is the circumcenter of ABC , then the value of 8(h + k) is equal to

Options

  1. A16
  2. B24
  3. C2
  4. D40

Correct answer

A. 16

Step-by-step solution

The slope of the given line 2x - y = 0 is m₀ = 2 . Let the slope of L₁ or L₂ be m . Since the angle between them is 4 , we have: ( 4 ) = | m - 2 1 + 2m | 1 = | m - 2 1 + 2m | This gives two equations: m - 2 = 1 + 2m m = -3 m - 2 = -(1 + 2m) 3m = 1 m = 1 3 The slopes of L₁ and L₂ are -3 and 1 3 . Since their product is (-3) ( 1 3 ) = -1 , the lines L₁ and L₂ are perpendicular. Thus, ABC is a right-angled triangle, with the right angle at A . In a right-angled triangle, the circumcenter is the midpoint of the hypoten

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