Two $⊥$ tangents to the circle $x^{2} + y^{2} = a^{2}$ meet at a point P. The locus of P has the equation

x^{2} + y^{2} = 3 a^{2}

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x^{2} + y^{2} = 2 a^{2}

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x^{2} + y^{2} = 4 a^{2}

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None of these

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Solution

The correct option is B $x^{2} + y^{2} = 2 a^{2}$
The coordinates of $P$ be $(h, k)$ . Then the equation of the tangents drawn from $P (h, k)$ to $x^{2} + y^{2} = a^{2}$ is
$(x^{2} + y^{2} - a^{2}) (h^{2} + k^{2} - a^{2}) = {(h x + h y - a^{2})}^{2}$ (using SS' $= T^{2}$ )
This equation represents a pair of perpendicular lines.
Therefore, coefficient of $x^{2}$ $+$ coefficient of $y^{2} = 0$
$\Rightarrow (h^{2} + k^{2} - a^{2} - h^{2}) + (h^{2} + k^{2} - a^{2} - k^{2}) = 0$
$\Rightarrow h^{2} + k^{2} = {2 a}^{2}$
Hence, locus is $x^{2} + y^{2} = 2 a^{2}$

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