First slide

Definition of a circle

Question

If the circle $x^{2} + y^{2} + 2 x + 3 y + 1 = 0$ cuts $x^{2} + y^{2} + 4 x + 3 y + 2 = 0$ in $A$ and $B,$ then the equation of the circle on $A B$ as diameter, is

Moderate

A

$x^{2} + y^{2} + x + 3 y + 3 = 0$
B

$2 x^{2} + 2 y^{2} + 2 x + 6 y + 1 = 0$
C

$x^{2} + y^{2} + x + 6 y + 1 = 0$
D

none of these

Solution

The equation of the common chord $A B$ of the two circles is
$2 x + 1 = 0$ [Using $S_{1} - S_{2} = 0$ ]
The equation of the required circle is
$(x^{2} + y^{2} + 2 x + 3 y + 1) + λ (2 x + 1) = 0$
[Using $: S_{1} + λ (S_{2} - S_{1}) = 0$ ]
$\Rightarrow x^{2} + y^{2} + 2 x (λ + 1) + 3 y + λ + 1 = 0 .$
Since $A B$ is a diameter of this circle. Therefore, centre of this circle lies on $A B .$
$∴ - 2 λ - 2 + 1 = 0 \Rightarrow λ = - 1 / 2$
So, the equation of the required circle is
$x^{2} + y^{2} + x + 3 y + 1 / 2 = 0$
or, $2 x^{2} + 2 y^{2} + 2 x + 6 y + 1 = 0$

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