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Direction Cosines

xy=a2 and x...

Question

$x y = a^{2}$ and $x^{2} + y^{2} = 2 a^{2}$
These two curves :

A

touch.

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B

intersect

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C

never intersect

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D

none of the above

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Solution

The correct option is A touch.
Touch: The two curves meet where $(x^{2} - a^{2}) = 0$
$∴ x = a, - a$ and so $y = \frac{a^{2}}{x} = a$ or $- a .$
Hence the points are $(a, a)$ or $(- a, - a) .$
$\frac{d y}{d x^{1}} = - \frac{y}{x} = - 1. {\frac{d y}{d x}}_{I I} = - \frac{x}{y} = - 1$
Hence the curves touch.

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