First slide

Definition of a circle

Question

The tangents drawn from the origin to the circle $x^{2} + y^{2} - 2 p x - 2 q y + q^{2} = 0$ are perpendicular if

Moderate

A

$p^{2} + q^{2} = 1$
B

$p^{2} - q^{2} = 1$
C

$p^{2} - q^{2} = 0$
D

none of these

Solution

Equation of the given circle can be written as
$(x - p)^{2} + (y - q)^{2} = p^{2}$ (i)
This has $(p, q)$ as the centre and $p$ as the radius showing that it touches $y -$ axis. So one of the tangents from the origin to the circle is $y -$ axis.
$\Rightarrow$ Other tangent from the origin to the circle must be $x -$ axis. which is possible if $q = \pm p$ .
$\Rightarrow p^{2} - q^{2} = 0 .$

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