First slide

Definition of a circle

Question

If a circle passes through $(1, 2)$ and cuts the circle $x^{2} + y^{2} = 4$ orthogonally then the equation of the locus of its centre, is

Moderate

A

$2 x + 4 y - 9 = 0$
B

$2 x + 4 y + 9 = 0$
C

$2 x - 4 y + 9 = 0$
D

none of these

Solution

Let the circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0 .$ This passes through $(1, 2)$ and cuts the circle $x^{2} + y^{2} = 4$ orthogonally.
$∴ 5 + 2 g + 4 f + c = 0 \dots (i)$ and, $0 = c - 4 . . . (i i)$
Eliminating $c$ from (i) and (ii), we get
$2 g + 4 f + 9 = 0 .$
Hence the locus of the centre $(- g, - f)$ is
$- 2 x - 4 y + 9 = 0$ or, $2 x + 4 y - 9 = 0$

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