Question

A circle is represented by the equation x2+y2−10x+4y=−13 . Find the center and the radius of the circle.

Type your answers in as numeral numbers only. Make sure to include the negative sign, if needed.

(3 points)
Center: (
,
)

Radius:

Answer 1

To find the center and radius of the circle represented by the equation \( x^2 + y^2 - 10x + 4y = -13 \), we need to rewrite it in standard form. The standard form of a circle is given by

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

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

First, let's rearrange the given equation:

\[
x^2 - 10x + y^2 + 4y = -13
\]

Next, we will complete the square for both \(x\) and \(y\).

**Completing the square for \(x\):**

1. Take the coefficient of \(x\), which is -10, halve it to get -5, and square it: \((-5)^2 = 25\).
2. Add and subtract 25 in the equation:

\[
x^2 - 10x + 25 - 25 + y^2 + 4y = -13
\]

**Completing the square for \(y\):**

1. Take the coefficient of \(y\), which is 4, halve it to get 2, and square it: \(2^2 = 4\).
2. Add and subtract 4 in the equation:

\[
x^2 - 10x + 25 + y^2 + 4y + 4 - 25 - 4 = -13
\]

Now we write the equation as:

\[
(x - 5)^2 + (y + 2)^2 = -13 + 25 + 4
\]

Calculating the right side:

\[
-13 + 25 + 4 = 16
\]

So, we have:

\[
(x - 5)^2 + (y + 2)^2 = 16
\]

Now we can identify the center and radius:

- Center \((h, k) = (5, -2)\)
- Radius \(r = \sqrt{16} = 4\)

Thus, the answers are:

**Center: (5, -2)**

**Radius: 4**