Question

Which equation represents a circle with a center at (–3, –5) and a radius of 6 units?

(x – 3)2 + (y – 5)2 = 6
(x – 3)2 + (y – 5)2 = 36
(x + 3)2 + (y + 5)2 = 6
(x + 3)2 + (y + 5)2 = 36

Answer 1

The standard equation of a circle with center at \((h, k)\) and radius \(r\) is given by:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

In this case, the center of the circle is \((-3, -5)\), so \(h = -3\) and \(k = -5\). The radius \(r\) is given as \(6\).

First, we need to calculate \(r^2\):

\[
r^2 = 6^2 = 36
\]

Substituting the values of \(h\), \(k\), and \(r^2\) into the standard form of the circle's equation, we have:

\[
(x - (-3))^2 + (y - (-5))^2 = 36
\]

This simplifies to:

\[
(x + 3)^2 + (y + 5)^2 = 36
\]

Thus, the equation that represents a circle with a center at \((-3, -5)\) and a radius of \(6\) units is:

\[
(x + 3)^2 + (y + 5)^2 = 36
\]

So the correct choice is:

\((x + 3)^2 + (y + 5)^2 = 36\)