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Thermodynamic Processes

If the parabo...

Question

If the parabola $y^{2} = 4 a (x - c_{1})$ and $x^{2} = 4 a (y - c_{2})$ touch each other, then the locus of their point of contact is

A

x y = 4 a^{2}

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B

x y = 2 a^{2}

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C

x y = a^{2}

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D

None of these

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Solution

The correct option is A $x y = 4 a^{2}$
Let $P (x, y)$ ne the point of contact.
$∴ 2 y \frac{d y}{d x} = 4 a$ and $2 x = 4 a \frac{d y}{d x}$

$\Rightarrow \frac{4 a}{2 y} = \frac{2 x}{4 a} \Rightarrow x y = 4 a^{2}$
which is the required locus.

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