Find the center and the radius of the circle $x^{2} + y^{2} + 8 x + 10 y - 8 = 0$

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Solution

Given equation of circle-
$x^{2} + y^{2} + 8 x + 10 y - 8 = 0 . . . . . (1)$
General form of equation of circle-
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0 . . . . . (2)$
Comparing equation $(1)$ and $(2)$ , we have
$g = 4$
$f = 5$
$c = - 8$
Therefore,
Radius of circle $= \sqrt{g^{2} + f^{2} - c} = \sqrt{4^{2} + 5^{2} - (- 8)} = \sqrt{16 + 25 + 8} = \sqrt{49} = 7$
Centre of circle $= (- g, - f) = (- 4, - 5)$

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