Question

Question 579
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 given the equation \( x^2 + y^2 - 16x - 10y + 40 = 0 \), we will complete the square for the \( x \) and \( y \) terms.

1. Rearrange the equation:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
2. Move the constant to the other side:
\[
x^2 - 16x + y^2 - 10y = -40
\]

3. Complete the square for the \( x \) terms:
\[
x^2 - 16x \rightarrow (x - 8)^2 - 64
\]
Here, we take half of \(-16\) which is \(-8\) and square it to get \(64\).

4. Complete the square for the \( y \) terms:
\[
y^2 - 10y \rightarrow (y - 5)^2 - 25
\]
Here, we take half of \(-10\) which is \(-5\) and square it to get \(25\).

5. Substitute the completed squares back into the equation:
\[
(x - 8)^2 - 64 + (y - 5)^2 - 25 = -40
\]
Simplifying this gives:
\[
(x - 8)^2 + (y - 5)^2 - 89 = -40
\]
So,
\[
(x - 8)^2 + (y - 5)^2 = 49
\]

6. The equation \((x - 8)^2 + (y - 5)^2 = 49\) represents a circle with center at \((8, 5)\) and radius \( r \) given by:
\[
r = \sqrt{49} = 7
\]

Thus, the radius of the circle is:
\[
\text{radius} = 7
\]