The tangents from which of the following points to the ellipse $5 x^{2} + 4 y^{2} = 20$ are perpendicular:

(\sqrt{5}, 2 \sqrt{2})

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(2 \sqrt{2}, 1)

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(\sqrt{5}, - 1)

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(\sqrt{5}, 1)

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Solution

The correct option is B $(2 \sqrt{2}, 1)$
The locus of the point of intersection of the tangents to the ellipse which meet at right angle, is a circle. This circle is called the "director circle" of the ellipse.
Points lies on director circle.
$x^{2} + y^{2} = a^{2} + b^{2} = 9$
Therefore, from the options $(2 \sqrt{2}, 1)$ lies on director circle.

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(4, \sqrt{2})

5 x^{2} + 4 y^{2} = 40.

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