Let P : y^ 2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of /4 with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is

JEE Main 2022 Mathematics question

Let P : , a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is

(1)8 only
(2)2 only
(3)1/4 only
(4)any a > 0Correct

Answer: (4)

Solution

A tangent y = mx + a/m makes an angle with the line of slope 3: = |(m − 3)/(1 + 3m)| = 1, so , giving m = −2 or m = 1/2. The point of contact of the tangent of slope m is

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Question ID VK-035235