Question

Calculate the center of the circle by completing the square of the equation x2+y2−10x+4y=20
𝑥
2
+
𝑦
2
−
10
𝑥
+
4
𝑦
=
20
.

Answer 1

To find the center of the circle by completing the square, we start with the given equation:

\[
x^2 + y^2 - 10x + 4y = 20
\]

We'll rearrange and isolate terms involving \(x\) and \(y\):

1. Rearranging the equation gives:
\[
x^2 - 10x + y^2 + 4y = 20
\]

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

**Completing the square for \(x\):**
- Take the coefficient of \(x\), which is -10.
- Divide it by 2: \(-10/2 = -5\).
- Square it: \((-5)^2 = 25\).
- Add and subtract this square inside the equation:

\[
x^2 - 10x + 25
\]

This becomes:

\[
(x - 5)^2
\]

**Completing the square for \(y\):**
- Take the coefficient of \(y\), which is 4.
- Divide it by 2: \(4/2 = 2\).
- Square it: \(2^2 = 4\).
- Add and subtract this square inside the equation:

\[
y^2 + 4y + 4
\]

This becomes:

\[
(y + 2)^2
\]

3. Now, add these squares to the original equation:

Starting from:

\[
(x - 5)^2 + (y + 2)^2 = 20 + 25 + 4
\]

4. Combine the constants on the right side:

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

Now, we can see that this is the standard form of the equation of a circle:

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

where \((h, k)\) is the center and \(r\) is the radius.

5. From \( (x - 5)^2 + (y + 2)^2 = 49\), we find:
- The center \((h, k)\) is \((5, -2)\).

Thus, the center of the circle is:

\[
\boxed{(5, -2)}
\]