TP and TQ are any two tangents to a parabola and the tangent at a third point R cuts them in `P` and `Q` , prove that `(TP')/(TP)+(TQ')/(TQ)=1`.

Options

A. 1
B. 2
C. 3
D. none of these

Answer

A

Explanation

Let parabola be `y^(2) = 4ax` and co-ordinates of P and Q on this parabola be `P -= (at_(1)^(2),2at_(1))` and `Q -= (at_(2)^(2),2at_(2))`,
T is the point of intersection of tangents at `t_(1)` and `t_(2)`. Co-ordinates of
`T -= {at_(1)t_(2),a(t_(1)+t_(2))}`
`P' -= {at_(1)t_(3),a(t_(1)+t_(3))}`
`Q' -= (at_(2)t_(3),a(t_(2)+t_(3))}`
Let `TP' : TP = lambda :1`
`:. lambda = (t_(3)-t_(2))/(t_(1)-t_(2))`
`rArr (TP')/(TP) = (t_(3)-t_(2))/(t_(1)-t_(2))`
Similarly, `(TQ')/(TQ) = (t_(1)-t_(3))/(t_(1)-t_2), (TP')/(TP) + (TQ')/(TQ) =1`.