Question

Calculate the radius of a circle by completing the square of the equation x2 + y2 −16x −10y +40 =0
𝑥
2

+

𝑦
2

−
16
𝑥

−
10
𝑦

+
40

=
0
.(1 point)
Responses

-7
-7

7
7

4
4

6

Answer 1

To find the radius of the circle given by the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\), we first need to rewrite the equation by completing the square for both \(x\) and \(y\).

Starting with the equation:

\[
x^2 + y^2 - 16x - 10y + 40 = 0
\]

We can rearrange it to isolate the constant on one side:

\[
x^2 - 16x + y^2 - 10y = -40
\]

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

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

1. Take the coefficient of \(x\) (which is -16), halve it to get -8, and then square it:

\[
(-8)^2 = 64
\]

2. Rewrite the \(x\) portion:

\[
x^2 - 16x = (x - 8)^2 - 64
\]

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

1. Take the coefficient of \(y\) (which is -10), halve it to get -5, and then square it:

\[
(-5)^2 = 25
\]

2. Rewrite the \(y\) portion:

\[
y^2 - 10y = (y - 5)^2 - 25
\]

Now, substitute back into the equation:

\[
(x - 8)^2 - 64 + (y - 5)^2 - 25 = -40
\]

Simplify the left side:

\[
(x - 8)^2 + (y - 5)^2 - 89 = -40
\]

Add 89 to both sides:

\[
(x - 8)^2 + (y - 5)^2 = 49
\]

This equation is now in the standard form of a circle's equation:

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

where \((h, k)\) is the center and \(r\) is the radius. From our completed square:

- The center is \((8, 5)\)
- The radius \(r\) is \(\sqrt{49} = 7\)

Therefore, the radius of the circle is:

\[
\boxed{7}
\]