To find the center and radius of the circle from the given equation, we first need to rewrite the equation in the standard form of a circle's equation:

$(x - h)^2 + (y - k)^2 = r^2$

The given equation is:

$x^2 + y^2 + 10x - 8y - 23 = 0$

Step 1: Group the $x$ and $y$ terms together:

$(x^2 + 10x) + (y^2 - 8y) = 23$

Step 2: Complete the square for the $x$ and $y$ terms.

• For the $x$ terms $x^2 + 10x$, take half of the coefficient of $x$ (which is 10), square it, and add it inside the square: $\left(\frac{10}{2}\right)^2 = 25$ So, add 25 to both sides: $(x^2 + 10x + 25)$

• For the $y$ terms $y^2 - 8y$, take half of the coefficient of $y$ (which is -8), square it, and add it inside the square: $\left(\frac{-8}{2}\right)^2 = 16$ So, add 16 to both sides: $(y^2 - 8y + 16)$

Now, the equation becomes:

$(x^2 + 10x + 25) + (y^2 - 8y + 16) = 23 + 25 + 16$

Step 3: Simplify the equation.

$(x + 5)^2 + (y - 4)^2 = 64$

Step 4: Identify the center and radius.

• The center $(h, k)$ is $(-5, 4)$.
• The radius $r$ is $\sqrt{64} = 8$.

So, the center is $(-5, 4)$, and the radius is $8$.

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How can you identify the center and radius directly from the general form of a circle’s equation?
2. What is the process of completing the square, and why is it used in this context?
3. How do changes in the coefficients of $x$ and $y$ affect the position of the circle?
4. What happens to the circle's equation if the circle is translated in the coordinate plane?
5. How would the circle's equation change if the circle were expanded or contracted?

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