Question

Find an equation for the circle that has center (−2, 4) and passes through the point (3, −4)

Answer 1

To find the equation of a circle, we use the standard form of the equation of a circle:

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

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

Given the center \((h, k) = (-2, 4)\), we can substitute these values into the equation:

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

Next, we need to find the radius \(r\). To do this, we calculate the distance between the center \((-2, 4)\) and the point \((3, -4)\) that the circle passes through, using the distance formula:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of the center \((-2, 4)\) as \((x_1, y_1)\) and the point \((3, -4)\) as \((x_2, y_2)\):

\[
r = \sqrt{(3 - (-2))^2 + (-4 - 4)^2}
\]
\[
= \sqrt{(3 + 2)^2 + (-4 - 4)^2}
\]
\[
= \sqrt{5^2 + (-8)^2}
\]
\[
= \sqrt{25 + 64}
\]
\[
= \sqrt{89}
\]

Now we have \(r^2 = 89\). We can substitute this back into the equation of the circle:

\[
(x + 2)^2 + (y - 4)^2 = 89
\]

Thus, the equation of the circle is:

\[
\boxed{(x + 2)^2 + (y - 4)^2 = 89}
\]