Question

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

Answer 1

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

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

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

Given the center \((-4, 6)\) and radius \(2\), you can identify \(h = -4\), \(k = 6\), and \(r = 2\).

Now, calculating \(r^2\):

\[
r^2 = 2^2 = 4
\]

Substituting these values into the equation of the circle gives:

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

This simplifies to:

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

Now, filling in the information, the equation that was requested in the format provided appears to ask for the transformation of equation:

\[
(x - 2) + (y - 2) = ?
\]

Since our derived equation is \((x + 4)^2 + (y - 6)^2 = 4\), we can re-express it in relation to the format, although it's not directly possible since it asks for a linear equation instead.

The correct answer to fill in the missing part (which asks for the right-hand side of the circle equation) is:

\[
4
\]

So the completed equation would be:

\[
(x + 4)^2 + (y - 6)^2 = 4
\] and the final answer is \(4\).