https://file.aiquickdraw.com/mathbot/file/1727989537761oixgmpsg.jpg

The problem in the image requires determining the center and the radius of the circle from the given equation:

\[
x^2 + y^2 - 10x - 8y - 23 = 0
\]

To solve this, we will complete the square for both the \(x\) and \(y\) terms to rewrite the equation in the standard form of a circle:

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

### Step 1: Group the \(x\) and \(y\) terms
\[
(x^2 - 10x) + (y^2 - 8y) = 23
\]
Move the constant \( -23 \) to the right-hand side.

### Step 2: Complete the square
For the \(x\)-terms:
\[
x^2 - 10x
\]
Take half of \(-10\), square it, and add to both sides. Half of \(-10\) is \(-5\), and \((-5)^2 = 25\).
\[
x^2 - 10x + 25
\]

For the \(y\)-terms:
\[
y^2 - 8y
\]
Take half of \(-8\), square it, and add to both sides. Half of \(-8\) is \(-4\), and \((-4)^2 = 16\).
\[
y^2 - 8y + 16
\]

So the equation becomes:
\[
(x^2 - 10x + 25) + (y^2 - 8y + 16) = 23 + 25 + 16
\]
\[
(x - 5)^2 + (y - 4)^2 = 64
\]

### Step 3: Identify the center and radius
Now the equation is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:
- \(h = 5\), \(k = 4\) is the center of the circle
- \(r^2 = 64\), so \(r = 8\) is the radius.

### Final Answer:
- **Center**: \((5, 4)\)
- **Radius**: \(8\)

Would you like any further explanation on completing the square or working with circle equations?

### Related Questions:
1. How do you complete the square for a quadratic expression?
2. What is the standard form of a circle’s equation, and how is it derived?
3. What is the significance of the radius and center in the equation of a circle?
4. How do you derive the formula for the radius from the general equation of a circle?
5. Can this technique of completing the square be used in other conic sections, such as ellipses?

### Tip:
Always remember that the method of completing the square is a key technique for transforming quadratic equations into standard forms, especially useful for conics like circles and parabolas.

Determine the coordinates of the center and the length of the radius of the circle from the equation: x^2 + y^2 - 10x - 8y - 23 = 0.

Learn how to determine the center and radius of a circle from the equation x^2 + y^2 - 10x - 8y - 23 = 0 by completing the square. This detailed guide covers key geometric concepts and is ideal for students in Grades 9-11.

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