Consider the parabola y 2 = 4 x . Let S be the focus of the parabola. A pair of tangents drawn to the parabola from the point P = - 2 , 1 meet the parabola at P 1 and P 2 . Let Q 1 and Q 2 be points on the lines S P 1 and S P 2 respectively such that P Q 1 is perpendicular to S P 1 and P Q 2 is perpendicular to S P 2 . Then, which of the following is/are TRUE?
D. S Q 2 = 1
On plotting the diagram of the given value we have, Now let P 1 t 2 , 2 t then tangent at P 1 will be t y = x + t 2 Since it passes through - 2 , 1 ⇒   t 2 - t - 2 = 0 ⇒ t = 2 , - 1 So, we get points P 1 4 , 4 and P 2 1 , - 2 Now finding the equation of S P 1 : 4 x - 3 y - 4 = 0 And equation of S P 2 : x - 1 = 0 Now finding the point by foot of point on line formula, Q 1 : x 1 + 2 4 = y 1 - 1 - 3 = - - 8 - 3 - 4 25 = 3 5 We get x 1 = 2 5 , y 1 = - 4 5 Similarly for Q 2 = 1 , 1 Now using the distance formula we get, S Q 1 = 1 - 2 5 2 + 4 5 2 = 1 Q 1 Q 2 = 9 25 + 81 25 = 90 25 = 3 10 5 P Q 1 = 144 25 + 81 25 = 3 S Q 2 = 1 .
Related: Mathematics — Parabola · All PYQ Banks