Question

Calculate the radius of a circle by completing the square of the equation x2 + y2 −16x −10y +40 =0 .(1 point) Responses 4 4 7 7 6 6 -7

Answer 1

To find the radius of the circle given by the equation \( x^2 + y^2 - 16x - 10y + 40 = 0 \), we first want to rearrange the equation into the standard form of a circle, which is \( (x - h)^2 + (y - k)^2 = r^2 \).

Let's start by moving the constant term to the other side of the equation:

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

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

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

1. From \( x^2 - 16x \):
- To complete the square, take half of -16, which is -8, and square it: \( (-8)^2 = 64 \).
- So, \( x^2 - 16x = (x - 8)^2 - 64 \).

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

2. From \( y^2 - 10y \):
- Take half of -10, which is -5, and square it: \( (-5)^2 = 25 \).
- So, \( y^2 - 10y = (y - 5)^2 - 25 \).

Now we can substitute back into our equation:

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

This simplifies to:

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

Adding 89 to both sides:

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

Now, we see that the equation of the circle is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:

- \( h = 8 \)
- \( k = 5 \)
- \( r^2 = 49 \)

To find the radius \( r \):

\[
r = \sqrt{49} = 7
\]

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