First slide

Analysis of two circles in circles

Question

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

Moderate

A

$2 a x - 2 b y + (a^{2} + b^{2} + 4) = 0$
B

$2 a x + 2 b y - (a^{2} + b^{2} + 4) = 0$
C

$2 a x + 2 b y + (a^{2} + b^{2} + 4) = 0$
D

$2 a x - 2 b y - (a^{2} + b^{2} + 4) = 0$

Solution

Let the equation of the circle be
$x^{2} + y^{2} - 2 g x - 2 f y + c = 0 .$ As it passes through $(a, b)$
$a^{2} + b^{2} - 2 g a - 2 f b + c = 0$
Since it intersects the circle $x^{2} + y^{2} = 4$
orthogonally
$2 g \times 0 + 2 f \times 0 = c - 4 \Rightarrow c = 4$
and the locus of $(g, f),$ the centre is
$2 a x + 2 b y - (a^{2} + b^{2} + 4) = 0$

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