Conic-Sections Question 564

Question: If the chords of contact of tangents from two points $ (\alpha ,\beta ) $ and $ (\gamma ,\delta ) $ to the ellipse $ \frac{x^{2}}{5}+\frac{y^{2}}{2}=1 $ are perpendicular, then $ \frac{\alpha \gamma }{\beta \delta }= $

Options:

A) $ \frac{4}{25} $

B) $ \frac{-4}{25} $

C) $ \frac{25}{4} $

D) $ \frac{-25}{4} $

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Answer:

Correct Answer: D

Solution:

[d] The equation of chord of contact of tangents form two points $ (\alpha ,\beta ) $ and $ (\gamma ,\delta ) $ to the given ellipse are $ \frac{x\alpha }{5}+\frac{y\beta }{2}=1 $ ? (1) and $ \frac{x\gamma }{5}+\frac{y\delta }{2}=1 $ ? (2) Since (1) and (2) are $ \bot $ ,
$ \therefore \frac{-2\alpha }{5\beta }\times \frac{-2\gamma }{5\delta }=-1\Rightarrow \frac{\alpha \gamma }{\beta \delta }=-\frac{25}{4} $