A parabola C with focus (0, 0) and passing through P(3,4). Equation of tangent at P to parabola is $3 x + 4 y - 25 = 0$ . A chord through S parallel to the tangent at P intersects the parabola at A and B. Which of the following statements is (are) CORRECT ?

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length of AB is 20 units

length of latus rectum of parabola is 20 units

only one real normal can be drawn from the point ( $- 3, - 4$ )

only one real normal can be drawn from the point ( $- 6, - 8$ )

answer is A, B, C.

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Detailed Solution

slope of $S P = \frac{4}{3}$
slope of tangent at P is $\frac{- 3}{4}$
$\Rightarrow P$ is vertex of parabola
Axis of parabola is 4x – 3y = 0
Question Image
AB = length of latus reaction = 4a
a = 5
AB = 20 units
LLR = 4a = 20 units

PM = 2a, then through M only one real normal can be drawn i,o., along axis of parabola