First slide

Analysis of two circles in circles

Question

The equation of a circle which passes through $(2 a, 0)$ and whose radical axis in relation to the circle $x^{2} + y^{2} = a^{2}$ is $x = a / 2,$ is

Moderate

A

$x^{2} + y^{2} - a x = 0$
B

$x^{2} + y^{2} + 2 a x = 0$
C

$x^{2} + y^{2} - 2 a x = 0$
D

$x^{2} + y^{2} + a x = 0$

Solution

The required circle is
$x^{2} + y^{2} - a^{2} + λ (x - \frac{a}{2}) = 0$ $[Using: S + λ L = 0]$
This passes through $(2 a, 0)$
$∴ 4 a^{2} - a^{2} + (\frac{3 a}{2}) λ = 0 \Rightarrow λ = - 2 a$
Hence, the required circle is
$\begin{array}{l} x^{2} + y^{2} - a^{2} - 2 a (x - \frac{a}{2}) = 0 \\ \Rightarrow x^{2} + y^{2} - a^{2} - 2 a x + a^{2} = 0 \Rightarrow x^{2} + y^{2} - 2 a x = 0 \end{array}$

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