In the parabola `y^2=4a x ,` then tangent at `P` whose abscissa is equal to the latus rectum meets its axis at `T ,` and normal `P` cuts the curve again at `Qdot` Show that `P T: P Q=4: 5.`

Options

A. `5:4`
B. `2:1`
C. `3:4`
D. `4:5`

Explanation

Let P be `(at^(2),2at)`.
Since `at^(2)=4a`, we have t = 2.
Tangent at P is `2y=x+4a`, which meets x-axis at T (-4a,0).
If coordinates of Q are `(at_(1)^(2),2at_(1))`, then `t_(1)=-t-(2)/(t)=-3`
`:. Q " is " (9a, - 6a)`
`:. (PQ)^(2)=125a^(2)` and `(PT)^(2)=80a^(2)`
`rArr PT : PQ= 4:5`.