If X is the foot of the directrix on the a parabola. PP' is a double ordinate of the curve and PX meets the curve again in Q. Then prove that PQ passes through the focus of the parabola.

Options

A. line
B. circle
C. parabola
D. none of these

Answer

A

Explanation

`X: (-a,0)`
`PX: y = (2at-0)/(at^(2) +a) (x+a)`
or `(1+t^(2))y = 2t (x+a)`
Solving with parabola `y^(2) =4ax`, we get
`(4t^(2)(x+a)^(2))/((1+t^(2))^(2)) =4ax`
or `t^(2) (x+a)^(2) = ax (1+t^(2))^(2)`
or `t^(2) x^(2) -(a+at^(4)) x+ at^(2) =0`
or `xt^(2) (x-at^(2)) -a (x-at^(2)) =0`
or `x = (a)/(t^(2))`
`:. Q: ((a)/(t^(2)),(2a)/(t))`
Thus `P' (at^(2)-2at)` and `Q ((a)/(t^(2)),(2a)/(t))`, are extremities of focal chord.
So, tangents at P' and Q intersect on directrix.