First slide

Analysis of two circles in circles

Question

Two equal circles with their centres on $x$ and $y$ axis will possess the radical axis in the following form

Moderate

A

$a x - b y - \frac{a^{2} + b^{2}}{4} = 0$
B

$2 g x - 2 f y + g^{2} - f^{2} = 0$
C

$g^{2} x + f^{2} y - g^{4} - f^{4} = 0$
D

$2 g^{2} x + 2 f^{2} y - g^{4} - f^{4} = 0$

Solution

Let $(x - a)^{2} + (y - 0)^{2} = r^{2}$ and $x^{2} + (y - b)^{2} = r^{2}$
be two equal circles with their centres on x and y axes respectively.
Then, their radical axis is given by
$- 2 a x + a^{2} + 2 b y - b^{2} = 0$
$\Rightarrow 2 a x - 2 b y + b^{2} - a^{2} = 0$
Clearly, is of the form $2 g x - 2 f y + g^{2} - f^{2} = 0$

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