First slide

Definition of a circle

Question

The locus of the centre of a circle touching the lines
$x + 2 y = 0$ and $x - 2 y = 0$ , is

Moderate

A

$x y = 0$
B

$x = 0$
C

$y = 0$
D

none of these

Solution

Let $(h, k)$ be the center and $r$ be the radius of the circle. As it touches the lines $x + 2 y = 0$ and $x - 2 y = 0 .$
$∴ |\frac{h + 2 k}{\sqrt{5}}| = |\frac{h - 2 k}{\sqrt{5}}| =$ Radius
$\begin{array}{ll} \Rightarrow h + 2 k = \pm (h - 2 k) \\ \Rightarrow k = 0 or, h = 0 \end{array}$
$\Rightarrow$ Locus of $(h, k)$ is $x = 0$ or, $y = 0$
Hence, the locus of (h, k) is $x y = 0$ .

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