Question

A circle is drawn on a coordinate plane with the center point at (−4,6)(−4,6) and a radius of 2. Derive the equation of the circle from the given information. Fill in the missing information in the following equation.(3 points) (x- )2 +(y-)2 =

Answer 1

To derive the equation of a circle given its center and radius, we can use the standard form of the equation of a circle, which is:

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

Here, \((h, k)\) is the center of the circle, and \(r\) is the radius.

From the information given:
- The center of the circle is at \((-4, 6)\). Thus, \(h = -4\) and \(k = 6\).
- The radius \(r = 2\).

Now, we need to calculate \(r^2\):
\[
r^2 = 2^2 = 4
\]

Now, substituting \(h\), \(k\), and \(r^2\) into the standard form equation, we get:

\[
(x - (-4))^2 + (y - 6)^2 = 4
\]

This simplifies to:

\[
(x + 4)^2 + (y - 6)^2 = 4
\]

Now, filling in the missing information in the equation you provided:

\((x - \mathbf{-4})^2 + (y - \mathbf{6})^2 = \mathbf{4}\)

So, the complete equation is:

\[
(x + 4)^2 + (y - 6)^2 = 4
\]