Question

Calculate the radius of a circle by completing the square of the equation x2+y2−16x−10y+40=0
.(1 point)
radius =

Answer 1

To find the radius of the circle defined by the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\), we will complete the square for both the \(x\) and \(y\) terms.

1. **Rearrange the equation**:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
Moving \(40\) to the other side gives us:
\[
x^2 - 16x + y^2 - 10y = -40
\]

2. **Complete the square** for \(x\):
- Take the coefficient of \(x\) (which is \(-16\)), halve it to get \(-8\), and square it to get \(64\).
- Thus, we can rewrite \(x^2 - 16x\) as:
\[
(x - 8)^2 - 64
\]

3. **Complete the square** for \(y\):
- Take the coefficient of \(y\) (which is \(-10\)), halve it to get \(-5\), and square it to get \(25\).
- Thus, we can rewrite \(y^2 - 10y\) as:
\[
(y - 5)^2 - 25
\]

4. **Rewriting the equation using completed squares**:
\[
(x - 8)^2 - 64 + (y - 5)^2 - 25 = -40
\]
Simplifying further gives:
\[
(x - 8)^2 + (y - 5)^2 - 89 = -40
\]
\[
(x - 8)^2 + (y - 5)^2 = 49
\]

5. **Identify the center and radius**:
The equation \((x - 8)^2 + (y - 5)^2 = 49\) is in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\) where \((h, k)\) is the center and \(r\) is the radius.

Here, the center is at \((8, 5)\) and \(r^2 = 49\). Therefore, the radius \(r\) is:
\[
r = \sqrt{49} = 7
\]

Thus, the radius of the circle is \(\boxed{7}\).