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JEE Main20264 April 2026Evening ShiftMathematicsParabolaActual

Let A,B and C be the vertices of a variable right angled triangle inscribed in the parabola y^2=16x . Let the vertex B containing the right angle be (4,8) and the locus of the centroid of ABC be a conic C_o . Then three times the length of latus rectum of C_o is ______

Correct answer

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Step-by-step solution

The equation of the parabola is y^2 = 16x , which gives a = 4 . Any point on the parabola can be taken as (4t^2, 8t) . The vertex B is given as (4,8) , which corresponds to the parameter t = 1 . Let the parameters for vertices A and C be t₁ and t₂ respectively. Thus, A (4t₁^2, 8t₁) and C (4t₂^2, 8t₂) . The slope of AB is: m_ AB = 8t₁ - 8 4t₁^2 - 4 = 2(t₁ - 1) (t₁ - 1)(t₁ + 1) = 2 t₁ + 1 Similarly, the slope of BC is: m_ BC = 2 t₂ + 1 Since ABC is right-angled at B , AB BC . Therefore, m_ AB m_ BC = -1 : 2 t₁ + 1 2

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