Answer
Equation of a circle with center $\left( -7,-4 \right)$ and radius $4$ is${{\left( x+7 \right)}^{2}}+{{\left( y+4 \right)}^{2}}=16$
Work Step by Step
Draw a circle with center $\left( -7,-4 \right)$ tangent to the y-axis.
From the figure we see that the circle touches the y-axis at $\left( -7,0 \right)$.
Since the radius is the distance between $\left( -7,0 \right)$ and $\left( -7,-4 \right)$.
$\sqrt{{{\left( -7-\left( -7 \right) \right)}^{2}}+{{\left( -4-\left( 0 \right) \right)}^{2}}}$
$r=4$
Put the value of radius $4$ and center coordinate$ \left( -7,-4 \right)$ in the standard equation of the circle:
$\begin{align}
& {{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}} \\
& {{\left( x-\left( -7 \right) \right)}^{2}}+{{\left( y-\left( -4 \right) \right)}^{2}}={{\left( 4 \right)}^{2}} \\
& {{\left( x+7 \right)}^{2}}+{{\left( y+4 \right)}^{2}}=16
\end{align}$
Thus, the equation of the circle with center $\left( -7,-4 \right)$ and radius $4$ is ${{\left( x+7 \right)}^{2}}+{{\left( y+4 \right)}^{2}}=16$.
