Explanation
Given equation: \(7\left(y^{2}+10 y\right)+7 x^{2}=1\)
\(7 y^{2}+70 y+7 x^{2}=1\)
We divide through by 7
\(y^{2}+10 y+x^{2}=1 / 7\)
we complete the square for \(y^{2}+10 y\) as follows - add
half the coefficient of \(y\left(\right.\) i.e. \(\left.\frac{10}{2}=5\right)\) and subtract the square of the result as shown
\begin{aligned}
(y+5)^{2}-5 z+x^{2}=1 / 7 \\
(y+5)^{2}+x^{2}=1 / 7+25 \\
(y+5)^{2}+(x+0)^{2} &=176 / 7 \\
(y+5)^{2}-5^{2}+x^{2} &=\frac{1}{7} \\
(1+5)^{2}+x^{2} &=\frac{1}{7}+25 \\
(y+5)^{2}+(x+0)^{2} &=\frac{176}{7} \\
(x+0)^{2}+(y+5)^{2} &=\frac{176}{7}
\end{aligned}
\(=\left(\sqrt{\frac{176}{7}}\right)^{2}\)
We compare with the equation of a circle with centre a. \(b\) and radius \(r\)
\begin{array}{l}
(x-a)^{2}+(y-b)^{2}=r^{2} \\
a=0, b=-5 \\
=(0,-5)
\end{array}