First slide

Definition of a circle

Question

The equation of the circle having its centre on the line $x - 2 y - 3 = 0$ and passing through the point of intersection of the circles
$x^{2} + y^{2} - 2 x - 4 y + 1 = 0$ and $x^{2} + y^{2} - 4 x - 2 y + 4 = 0$ is

Moderate

A

$x^{2} + y^{2} - 6 x + 7 = 0$
B

$x^{2} + y^{2} - 3 x + 4 = 0$
C

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

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

Solution

The equation of a circle passing through tr-.:intersection
of the two given circles is
$(x^{2} + y^{2} - 2 x - 4 y + 1) + λ (x^{2} + y^{2} - 4 x - 2 y + 4) = 0$
$\Rightarrow x^{2} + y^{2} - 2 x (\frac{1 + 2 λ}{1 + λ}) - 2 y \frac{(2 + λ)}{1 + λ} + (\frac{1 + 4 λ}{1 + λ}) \div 0 \dots$ (i)
The co-ordinates of its centre are $(\frac{1 + 2 λ}{1 + λ}, \frac{2 + λ}{1 + λ})$
Since the centre lies on $x + 2 y - 3 = 0 .$
$∴ 1 + 2 λ + 4 + 2 λ - 3 - 3 λ = 0 \Rightarrow λ = - 2$
Putting $λ = - 2$ in (i), we obtain that the required circle is
$x^{2} + y^{2} - 6 x + 7 = 0 .$

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