First slide

Definition of a circle

Question

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

Moderate

A

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

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

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

none of these

Solution

Let 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
$(x^{2} + y^{2} - a^{2}) (h^{2} + k^{2} - a^{2}) = {(h x + k y - a^{2})}^{2}$ [Using $S S' = T^{2}$ ]
This equation represents a pair of perpendicular lines.
$Coeff. of x^{2} + Coeff. of y^{2} = 0$
$\Rightarrow (h^{2} + k^{2} - a^{2} - h^{2}) + (h^{2} + k^{2} - a^{2} - k^{2}) = 0$
$\Rightarrow h^{2} + k^{2} = 2 a^{2}$
Hence, the locus of $(h, k)$ is $x^{2} + y^{2} = 2 a^{2} .$

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