Question

find the center and radius of the circle with the given equation:

x^2 + y^2 - 12x + 2y + 21 = 0

Please show detailed step-by-step directions so I can figure out how to do this on my own, Thank you!

Answer 1

Given,

Equation is $x^{2}+y^{2}-12x+2y+21=0$

General equation of circle is $x^{2}+y^{2}+2gx+2fy+c=0$

Center is $C=(-g,-f)$

Radius is $r=\sqrt{g^{2}+f^{2}-c}$

Now comparing $x^{2}+y^{2}-12x+2y+21=0$ with $x^{2}+y^{2}+2gx+2fy+c=0$

Here,

$2g=-12$

$g=-6$

$2f=2$

$f=1$

Now Center C will be, $C=(-g,-f)=(6,-1)$

Radius will be,

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

$r=\sqrt{(-6)^{2}+1^{2}-21}$

$r=\sqrt{36+1-21}$

$r=\sqrt{37-21}$

$r=\sqrt{16}$

$r=4$

Therefore,

$Center=(6,-1)$

Radius = 4 units