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 the circle, we need to use the standard form of a circle's equation, which is given by:

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

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

### Step 1: Identify the center
The center of the circle is given as \((-2, 4)\). Therefore, we have:
- \(h = -2\)
- \(k = 4\)

### Step 2: Calculate the radius
The radius \(r\) can be calculated using the distance formula between the center and a point on the circle, which in this case is the point \((3, -4)\).

The distance formula is:

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

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

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

### Step 3: Write the equation of the circle
Now we can substitute \(h\), \(k\), and \(r^2\) into the standard form of the circle's equation:

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

This simplifies to:

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

### Final Result
The equation of the circle is:

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