Length of the chord of contact of (2,5) with respect to $y^{2} = 8 x$ is

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Solution

$\begin{matrix} U s e t h e p a r a m e t i c f o r m o f t h e p a r a b o l a x = a t^{2}, y = 2 a t w h e r e a = 2 x = 2 t^{2}, y = u t T h e e q u a t i o n o f tan g e n t a t (2 t^{2}, u t) i s t y = x + 2 t^{2} I t p a s s e s t h r o u g h (2, 5) ∴ 9 t = 2 + 2 t^{2} 2 t^{2} - 5 t + 2 = 0 2 t^{2} - u t - t + 2 = 0 2 t (t - 2) - 1 (t - 2) = 0 t_{1} = 2 o r t_{2} = \frac{1}{2} ∴ T h e l e n g t h o f t h e c h o r d = \sqrt{({t_{1}}^{2} - {t_{2}}^{2}) + {(3 t_{1} - 2 t_{2})}^{2}} = \sqrt{{(4 - \frac{1}{4})}^{2} + {(4 - 1)}^{2}} = \sqrt{{(\frac{15}{2})}^{2} + 3^{2}} = \sqrt{\frac{225 + 144}{16}} = \frac{\sqrt{369}}{4} = \frac{3 \sqrt{41}}{4} \end{matrix}$

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