Answer
$(x+5)^{2}+(y+1)^{2}=26$
Work Step by Step
The circle has center $(-5,-1)$ and passes through the origin.
Since the circle passes through the origin, the distance between the origin and the center of the circle must represent the radius.
Find the distance between the center of the circle and the origin. $x_{1}=-5$, $y_{1}=-1$, $x_{2}=0$ and $y_{2}=0$:
$d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
$d=\sqrt{(0+5)^{2}+(0+1)^{2}}=\sqrt{5^{2}+1^{2}}=\sqrt{25+1}=...$
$...=\sqrt{26}$
The radius of the circle is $r=\sqrt{26}$
Now, the center and the radius of the circle are known. Substitute them into the formula for the equation of a circle and simplify:
$(x-h)^{2}+(y-k)^{2}=r^{2}$
$(x+5)^{2}+(y+1)^{2}=26$
