Answer
Center of circle is $\left( 0,-5 \right)$ and radius is $r=10$
Work Step by Step
Standard equation of the circle is:
${{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}}$ (equation - 1)
And equation of circle is ${{x}^{2}}+{{y}^{2}}+10y=75$ (equation - 2)
Now add $25$ on both the sides of (equation - 2) to complete the square twice.
$\begin{align}
& {{x}^{2}}+{{y}^{2}}+10y+25=75+25 \\
& \left( {{x}^{2}} \right)+\left( {{y}^{2}}+10y+25 \right)=100 \\
& {{\left( x \right)}^{2}}+{{\left( y+5 \right)}^{2}}={{\left( 10 \right)}^{2}}
\end{align}$
Now compare the standard equation with the equation${{\left( x \right)}^{2}}+{{\left( y+5 \right)}^{2}}={{\left( 10 \right)}^{2}}$.
Center coordinate of circle is $\left( h=0,k=-5 \right)$.
And radius of circle is $r=10$.
To graph, we plot the points $\left( 0,5 \right)$, $\left( 0,-15 \right)$, $\left( -10,-5 \right)$, and $\left( 10,-5 \right)$ which are, respectively, $10$ units above, below, left and right of $\left( 0,-5 \right)$.
