Question

Calculate the center of the circle by completing the square of the equation x2+y2+2x−20y+5=0
𝑥
2
+
𝑦
2
+
2
𝑥
−
20
𝑦
+
5
=
0
.

Answer 1

To find the center of the circle given by the equation

\[
x^2 + y^2 + 2x - 20y + 5 = 0,
\]

we need to complete the square for both \(x\) and \(y\).

### Step 1: Rearranging the equation

First, we can move the constant term to the other side:

\[
x^2 + 2x + y^2 - 20y + 5 = 0 \implies x^2 + 2x + y^2 - 20y = -5.
\]

### Step 2: Completing the square for \(x\)

For the \(x\) terms \(x^2 + 2x\):

1. Take the coefficient of \(x\) (which is 2), divide by 2 to get 1, and square it (which gives \(1^2 = 1\)).
2. Add this square inside the equation and balance it by subtracting the same value.

So we have:

\[
x^2 + 2x = (x + 1)^2 - 1.
\]

### Step 3: Completing the square for \(y\)

For the \(y\) terms \(y^2 - 20y\):

1. Take the coefficient of \(y\) (which is -20), divide by 2 to get -10, and square it (which gives \((-10)^2 = 100\)).
2. Add this square inside the equation and balance it by subtracting the same value.

So we have:

\[
y^2 - 20y = (y - 10)^2 - 100.
\]

### Step 4: Substitute back into the equation

Now we substitute these completed squares back into the equation:

\[
(x + 1)^2 - 1 + (y - 10)^2 - 100 = -5.
\]

This simplifies to:

\[
(x + 1)^2 + (y - 10)^2 - 101 = -5.
\]

Adding 101 to both sides gives:

\[
(x + 1)^2 + (y - 10)^2 = 96.
\]

### Step 5: Identifying the center and radius

Now, the equation is in the standard form of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

From the equation:

\[
(x + 1)^2 + (y - 10)^2 = 96,
\]

we see:

- The center \((h, k)\) is \((-1, 10)\).
- The radius is \(\sqrt{96}\).

### Conclusion

The center of the circle is at

\[
\boxed{(-1, 10)}.
\]