Question

A circle is represented by the equation x2+y2−10x+4y=−13 . Complete the square to discover 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.

Answer 1

To complete the square for the equation of the circle \(x^2 + y^2 - 10x + 4y = -13\), we start by rearranging it:

1. Rearrange the equation:
\[
x^2 - 10x + y^2 + 4y = -13
\]

2. Complete the square for the \(x\) terms:
- Take the coefficient of \(x\), which is \(-10\), halve it to get \(-5\), and square it to get \(25\).
- Add and subtract \(25\) to the equation.

3. Complete the square for the \(y\) terms:
- Take the coefficient of \(y\), which is \(4\), halve it to get \(2\), and square it to get \(4\).
- Add and subtract \(4\) to the equation.

Now, the equation becomes:
\[
(x^2 - 10x + 25) + (y^2 + 4y + 4) = -13 + 25 + 4
\]

This simplifies to:
\[
(x - 5)^2 + (y + 2)^2 = 16
\]

Now, we can identify the center and the radius:
- The center of the circle is \((5, -2)\).
- The radius is \(\sqrt{16} = 4\).

Thus, the answers are:
- Center: \(5, -2\)
- Radius: \(4\)

So, the final answer is:
**Center: 5, -2; Radius: 4**