Question

write the standard form of the equation of the circle with the given center and radii (0, - 2); $sqrt{19}$

a. $x^{2}+(y + 2)^{2}=19$
b. $x^{2}+(y - 2)^{2}=19$
c. $(x - 2)^{2}+y^{2}=361$
d. $(x + 2)^{2}+y^{2}=361$

Explanation:

Step1: Recall circle - equation formula

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

Step2: Identify values of $h$, $k$, and $r$

Given the center $(0,-2)$ and radius $r = \sqrt{19}$, we have $h = 0$, $k=-2$, and $r=\sqrt{19}$.

Step3: Substitute values into the formula

Substitute $h = 0$, $k=-2$, and $r=\sqrt{19}$ into $(x - h)^2+(y - k)^2=r^2$. We get $(x - 0)^2+(y-(-2))^2=(\sqrt{19})^2$.

Step4: Simplify the equation

$(x - 0)^2=x^2$, $(y + 2)^2$, and $(\sqrt{19})^2 = 19$. So the equation is $x^2+(y + 2)^2=19$.

Answer:

A. $x^2+(y + 2)^2=19$