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Tangent To a Parabola

If the tangen...

Question

If the tangent at $P$ to the curve $x y = c^{2}$ meets the axes at $A, B$ then $P$ divides $A B$ in the ratio

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Solution

Accordingtoquestion................
$\begin{matrix} G i v e n p o i n t, o n t h e c u r v e x y = c^{2}, a s P = (c t, \frac{c}{t}) e q u a t i o n o f t a n g e n t a t c u r v e i s T = 0 ∴ \frac{x . \frac{c}{t} + y . c t}{2} = \frac{x}{t} + y t = 2 c N o w, I n x - a x i s w h e r e y = 0 s o, x = 2 c t, t h e n P o i n t A = (2 c t, 0) A n d, I n y - a x i s w h e r e x = o s o, y = \frac{2 c}{t}, t h e n P o i n t B = (0, \frac{2 c}{t}) N o w, T h e P o i n t P (c t, \frac{c}{t}) d i v i d e s t h e l i n e A B i n t h e r a t i o k : 1 s o, c t = \frac{k .0 + 1 \times 2 c t}{k + 1} \Rightarrow k c t + c t = 2 c t \Rightarrow k = 1 s o t h e r a t i o i s 1 : 1. A n s w e r . \end{matrix}$

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