What is the equation of the circle with center left parenthesis negative 4 comma negative 3 right parenthesis that passes through the point left parenthesis 6 comma 2 right parenthesis?
A. left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25
B. left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 125
C. left parenthesis x plus 4 right parenthesis squared plus left parenthesis y plus 3 right parenthesis squared equals 25
D. left parenthesis x plus 4 right parenthesis squared plus left parenthesis y plus 3 right parenthesis squared equals 125
The standard form equation for a circle with center $(h,k)$ and radius $r$ is $(x-h)^2+(y-k)^2=r^2$. We are given the center of the circle as $(-4,-3)$, so the equation starts as $(x+4)^2+(y+3)^2=r^2$. We just need to find the value of $r$, which is the distance from the center to the point $(6,2)$ on the circle. Using the distance formula, we have:
$\sqrt{(6-(-4))^2+(2-(-3))^2}=\sqrt{10^2+5^2}=\sqrt{125}=5\sqrt{5}$
So, the final equation of the circle is $(x+4)^2+(y+3)^2=(5\sqrt{5})^2=125$.
Therefore, the answer is $\boxed{\textbf{(D) } \left(x+4\right)^2+\left(y+3\right)^2 = 125}$.
To find the equation of a circle, you can use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle, and r represents the radius.
Given that the center is (-4, -3) and the circle passes through the point (6, 2), we can substitute these values into the formula:
(x - (-4))^2 + (y - (-3))^2 = r^2
(x + 4)^2 + (y + 3)^2 = r^2
Therefore, the correct equation of the circle is:
(x + 4)^2 + (y + 3)^2 = r^2
Out of the given options, the equation that matches this form is C. (x + 4)^2 + (y + 3)^2 = 25