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JEE Advanced2024MathematicsParabolaActual

Let A₁, B₁, C₁ be three points in the x y -plane. Suppose that the lines A₁ C₁ and B₁ C₁ are tangents to the curve y^2=8 x at A₁ and B₁ , respectively. If O=(0,0) and C₁=(-4,0) , then which of the following statements is (are) TRUE?

Options

  1. AThe length of the line segment O A₁ is 4 3
  2. BThe length of the line segment A₁ B₁ is 16
  3. CThe orthocenter of the triangle A₁ B₁ C₁ is (0,0)
  4. DThe orthocenter of the triangle A₁ B₁ C₁ is (1,0)

Correct answer

A. The length of the line segment O A₁ is 4 3

Step-by-step solution

Equation of tangent at (2 t^2, 4 t ) is ty = x +2 t ^2 It is passing through (-4,0) 0=-4+2 t ^2 t = 2 . array rl A ₁=(4,4 2 ) B ₁=(4,-4 2 ) array aligned & OA ₁= 48 =4 3 & ~A ₁ ~B ₁=8 2 aligned Equation of altitude of A₁ B₁ C₁ drawn from A₁ is y-4 2 = 2 (x-4) 2 x-y=0 ...(1) Equation of altitude of A₁ B₁ C₁ drawn from C₁ is x =0 ...(2) Solving (1) and (2) orthocentre is (0,0) correct options are (1), (3)

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