Find the center of this equation of a circle x^2+y^2−2x−2y−23=0


To find the center of the circle given by the equation \(x^2 + y^2 - 2x - 2y - 23 = 0\), follow these steps:

### Step 1: Rewrite the equation in standard form
The equation of a circle is typically expressed in the standard form:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center, and \(r\) is the radius.

### Step 2: Complete the square for both \(x\) and \(y\)
Start by grouping the \(x\) and \(y\) terms together:

\[
x^2 - 2x + y^2 - 2y = 23
\]

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

#### Completing the square for \(x\):
- Take the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\).
- Add and subtract \(1\) inside the equation.

\[
x^2 - 2x + 1 - 1
\]

This can be rewritten as:

\[
(x - 1)^2 - 1
\]

#### Completing the square for \(y\):
- Similarly, take the coefficient of \(y\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\).
- Add and subtract \(1\) inside the equation.

\[
y^2 - 2y + 1 - 1
\]

This can be rewritten as:

\[
(y - 1)^2 - 1
\]

### Step 3: Combine the completed squares
Now, substitute these back into the original equation:

\[
(x - 1)^2 - 1 + (y - 1)^2 - 1 = 23
\]

Simplify by moving the \(-1 - 1\) to the other side of the equation:

\[
(x - 1)^2 + (y - 1)^2 = 23 + 2
\]

\[
(x - 1)^2 + (y - 1)^2 = 25
\]

### Step 4: Identify the center and radius
- The center \((h, k)\) of the circle is \((1, 1)\).
- The radius \(r\) of the circle is \(\sqrt{25} = 5\).

So, the center of the circle is \((1, 1)\).

Would you like more details or have any questions? Here are some related questions:

1. How can you find the radius of the circle given its equation?
2. What is the geometric meaning of completing the square in a circle's equation?
3. How do you find the points of intersection between two circles?
4. Can you derive the equation of a circle given three points on its circumference?
5. How does the equation of a circle change if the circle is translated on the coordinate plane?

**Tip:** Completing the square is a useful technique not only for circles but also for solving quadratic equations and analyzing parabolas.

Learn how to find the center of a circle given its equation x^2 + y^2 - 2x - 2y - 23 = 0. Follow step-by-step instructions to complete the square and identify the center and radius of the circle. Suitable for high school students studying circle equations and completing the square.

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