If the circle $x^{2} + y^{2} + 2 a x + 2 b y + c = 0$ , then find the centre and the radius the circle.

(- a, - b), \sqrt{a^{2} + b^{2} - c}

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(- a, b), \sqrt{a^{2} + b^{2} + c}

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(a, - b), \sqrt{a^{2} + b^{2} - c}

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(a, b), \sqrt{a^{2} + b^{2} + c}

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Solution

The correct option is A $(- a, - b), \sqrt{a^{2} + b^{2} - c}$
We have,
$x^{2} + y^{2} + 2 a x + 2 b y + c = 0$ $. . . . . . . . . . . (1)$

On comparing the above equation with general equation
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

Then,
$2 g = 2 a \Rightarrow g = a$
$2 f = 2 b \Rightarrow f = b$
$c = c$

Therefore, the centre of the circle
$(- g, - f) = (- a, - b)$

And the radius of the circle
$r = \sqrt{g^{2} + f^{2} - c}$
$r = \sqrt{a^{2} + b^{2} - c}$

Hence, this is the answer.

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