The given equation of the circle is:

$x^2 + 6x + y^2 - 8y = 0$

To find the center and the radius of the circle, we will rewrite this equation in standard form by completing the square for both $x$ and $y$ terms.

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

$(x^2 + 6x) + (y^2 - 8y) = 0$

Step 2: Complete the square

• For the $x$-terms: $x^2 + 6x$ Take half of the coefficient of $x$ (which is 6), square it, and add it inside the parentheses. Half of 6 is 3, and $3^2 = 9$. So, add and subtract 9: $(x^2 + 6x + 9 - 9) = (x + 3)^2 - 9$

• For the $y$-terms: $y^2 - 8y$ Take half of the coefficient of $y$ (which is -8), square it, and add it inside the parentheses. Half of -8 is -4, and $(-4)^2 = 16$. So, add and subtract 16: $(y^2 - 8y + 16 - 16) = (y - 4)^2 - 16$

Step 3: Rewrite the equation

Now, substitute the completed squares back into the equation: $(x + 3)^2 - 9 + (y - 4)^2 - 16 = 0$

Step 4: Simplify

Combine constants on the right-hand side: $(x + 3)^2 + (y - 4)^2 = 25$

Step 5: Interpret the standard form

The equation is now in the standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$ Where $(h, k)$ is the center and $r$ is the radius.

From the equation $(x + 3)^2 + (y - 4)^2 = 25$, we can see that:

• The center of the circle is $(-3, 4)$.
• The radius is $\sqrt{25} = 5$.

Final Answer:

• Center: $(-3, 4)$
• Radius: $5$

Would you like any further details or explanations?

Here are five related questions to expand your understanding:

1. How do you complete the square for a general quadratic equation?
2. What is the geometric interpretation of the equation of a circle?
3. How do you convert an ellipse equation into standard form?
4. What is the general formula for the equation of a circle in 3D?
5. How does the radius of a circle change if you alter its equation?

Tip: When completing the square, always remember to balance both sides of the equation by adding or subtracting constants accordingly.