JEE Main · MathematicsHard

$A variable chord of the parabola y^{2} = 4 a x subtends a right angle at the vertex. If the chord passes through a fixed point (h, 0), find the value of h .$

Show correct answer & step-by-step solution

Correct answer: C — $4 a$

Solution

$Let the endpoints of the chord be (a t_{1}^{2}, 2 a t_{1}) and (a t_{2}^{2}, 2 a t_{2}) .$
$The slopes of the lines to the origin are m_{1} = 2/ t_{1} and m_{2} = 2/ t_{2} .$
$Since the angle is 90 degrees, m_{1} m_{2} = - 1, implying t_{1} t_{2} = - 4 .$
$The equation of the chord is y (t_{1} + t_{2}) = 2 x + 2 a t_{1} t_{2} .$
$Substituting t_{1} t_{2} = - 4 and y = 0 gives $0 = 2x - 8a$, so x = 4 a .$
$Hence the answer is (C).$