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Let the area of the triangle formed by the origin and the endpoints of a focal chord of the parabola y^2=16x be 48 square units. The length of this focal chord is _______

Correct answer

36

Step-by-step solution

The equation of the parabola is y^2 = 16x , so a = 4 . Let the endpoints of the focal chord be P(at^2, 2at) and Q ( a t^2 , - 2a t ) . The area of the triangle formed by the origin O(0,0) and the points P and Q is given by: Area = 1 2 |x₁ y₂ - x₂ y₁| Area = 1 2 | at^2 (- 2a t ) - ( a t^2 )(2at) | Area = 1 2 | -2a^2 t - 2a^2 t | = a^2 | t + 1 t | We are given that the area is 48 square units and a = 4 : 4^2 | t + 1 t | = 48 16 | t + 1 t | = 48 | t + 1 t | = 3 The length L of the focal chord with endpoints P(at^2, 2a

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