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

Given Equation:

$x^2 + y^2 + 12x - 8y - 29 = 0$

Step 1: Rearrange terms

Group the $x$-terms and $y$-terms together: $(x^2 + 12x) + (y^2 - 8y) = 29$

Step 2: Complete the square

• For the $x$-terms: $x^2 + 12x$

  • Take half the coefficient of $x$, square it: $(\frac{12}{2})^2 = 36$
  • Add and subtract $36$: $x^2 + 12x = (x + 6)^2 - 36$

• For the $y$-terms: $y^2 - 8y$

  • Take half the coefficient of $y$, square it: $(\frac{-8}{2})^2 = 16$
  • Add and subtract $16$: $y^2 - 8y = (y - 4)^2 - 16$

Step 3: Rewrite the equation

Substitute the completed squares back into the equation: $(x + 6)^2 - 36 + (y - 4)^2 - 16 = 29$

Combine constants: $(x + 6)^2 + (y - 4)^2 = 81$

Step 4: Write in standard form

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

From the equation: $(x + 6)^2 + (y - 4)^2 = 81$

• Center: $(-6, 4)$
• Radius: $r = \sqrt{81} = 9$

Final Answer:

• Center: $(x, y) = (-6, 4)$
• Radius: $r = 9$

Would you like a detailed step-by-step breakdown of completing the square or have any further questions?

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Here are 5 related questions:

1. How do you complete the square for a quadratic equation?
2. What is the geometric significance of the circle's equation in standard form?
3. How do you determine the distance between two points in a plane using their coordinates?
4. How would the equation of the circle change if the radius doubled?
5. Can you generalize the equation of a circle given its center and radius?

Tip: Always double-check your arithmetic when completing the square to avoid mistakes!