Question

write the standard form of the equation and the general form of the equation of the circle with radius r and center (h,k). then graph the circle. r = 4; (h,k)=(0, - 4)

Explanation:

Step1: Recall standard - form formula

The standard - form of the equation of a circle with radius $r$ and center $(h,k)$ is $(x - h)^2+(y - k)^2=r^2$.

Step2: Substitute given values

Substitute $r = 4$, $h = 0$, and $k=-4$ into the formula: $(x - 0)^2+(y+4)^2=4^2$, which simplifies to $x^{2}+(y + 4)^{2}=16$.

Step3: Expand to get general form

Expand the standard - form equation:
\[

$$\begin{align*} x^{2}+(y + 4)^{2}&=16\\ x^{2}+y^{2}+8y+16&=16\\ x^{2}+y^{2}+8y&=0 \end{align*}$$

\]

Answer:

Standard form: $x^{2}+(y + 4)^{2}=16$
General form: $x^{2}+y^{2}+8y = 0$