First slide

Analysis of two circles in circles

Question

The equation of the circle on the common chord of the circles
$(x - a)^{2} + y^{2} = a^{2}$ and $x^{2} + (y + b)^{2} = b^{2}$ as diameter is ,

Moderate

A

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

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

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

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

Solution

The equation of the common chord of the circles
$(x - a)^{2} + y^{2} = a^{2}$ and $x^{2} + (y + b)^{2} = b^{2}$
$a x + b y = 0$
The equation of the required circle is
$\{(x - a)^{2} + y^{2} - a^{2}\} + λ (a x + b y) = 0$
$\Rightarrow x^{2} + y^{2} + x (a λ - 2 a) + λ b y = 0$ (i)
Since (i) is a diameter of circle (ii).
$∴ a \{- \frac{(a λ - 2 a)}{2}\} + b (- \frac{λ, b}{2}) = 0 \Rightarrow λ = \frac{2 a^{2}}{a^{2} + b^{2}}$
Putting the value of A in (ii), the equation of the required circle is
$(a^{2} + b^{2}) (x^{2} + y^{2}) = 2 a b (b x - a y)$

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