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 :
$\frac{T P^{'}}{T P} + \frac{T Q^{'}}{T Q} = 1$ ,

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Solution

TP and TQ are two tangents to a parabola and the tangent at a third point R cuts them in $P^{'}$ and $Q^{'}$ .
Then prove that $\frac{T P^{'}}{T P} + \frac{T Q^{'}}{T Q} = 1$
Let
$P = (a t_{1}^{2}, 2 a t_{1}) Q = (a t_{2}^{2}, 2 a t_{2}) T = [a t_{1} t_{2}, a (t_{1} + t_{2})] R = (a t_{3}^{2}, 2 a t_{3}) P^{'} = [a t_{3} t_{1}, a (t_{3} + t_{1})] Q^{'} = [a t_{3} t_{2}, a (t_{3} + t_{2})]$
Let
$\frac{T P^{'}}{T P} = \frac{λ}{1} λ = \frac{t_{3} - t_{22}}{t_{1} - t_{2}} = \frac{T P^{'}}{T P} . . . . . . . . (1)$
Similarly $\frac{T Q^{'}}{T Q} = \frac{t_{1} - t_{3}}{t_{1} - t_{2}} . . . . . . . . . . . (2)$
$(1) + (2) \frac{T P^{'}}{T P} + \frac{T Q^{'}}{T Q} = 1$
Hence proved.

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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,

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 :
$\frac{T P^{'}}{T P} + \frac{T Q^{'}}{T Q} = 1$ ,