The radius of the circle S is same as the radius of $x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ and the centre of S is the centre of $x^{2} + y^{2} - 2 x - 4 y + 11 = 0$ . Find the equation of S.

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Solution

The correct option is C

To find the equation of a circle, we need the centre and radius of the circle, In this problem, we are given two circles, using which we can find the centre and radius.
To find the radius
$x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ and S(our circle) has the same radius.
We know the radius of the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is

$\sqrt{g^{2} + f^{2} - c}$

$\Rightarrow r = \sqrt{1^{2} + 2^{2} + 11}$

=4

To find the centre:
We are given S and $x^{2} + y^{2} - 2 x - 4 y + 11 = 0$ have the same centre
Centre of $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is $(- g, - f)$ .
$\Rightarrow$ centre of $x^{2} + y^{2} - 2 x - 4 y + 11 = 0$ is $(1, 2)$
We got the centre $(1, 2)$ and radius (4) of the circle.
$\Rightarrow$ Equation is $(x - 1)^{2} + (y - 2)^{2} = 4^{2}$

$\Rightarrow x^{2} + y^{2} - 2 x - 4 y + 1 + 4 = 16$

$\Rightarrow$ $x^{2} + y^{2} - 2 x - 4 y - 11 = 0$

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