The given equation appears to have a typo or some missing components since it is written as $x^2y^2 + 12x - 6y - 7 = 0$, which does not correspond to the standard form of any conic section. I will assume you meant to write a standard form equation of a circle.

If the intended equation is a circle, the correct form should be:

$x^2 + y^2 + 12x - 6y - 7 = 0.$

Step-by-Step Solution

To find the center and radius of the circle from this equation, we need to rewrite it in the standard form:

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

where $(h, k)$ is the center and $r$ is the radius. To do this, we'll complete the square for both $x$ and $y$.

1. Rearrange the equation

Let's group the $x$ and $y$ terms together:

$x^2 + 12x + y^2 - 6y = 7.$

2. Complete the square for $x$ and $y$

To complete the square:

• For $x^2 + 12x$: $x^2 + 12x = (x^2 + 12x + 36) - 36 = (x + 6)^2 - 36.$

• For $y^2 - 6y$: $y^2 - 6y = (y^2 - 6y + 9) - 9 = (y - 3)^2 - 9.$

Now substitute these back into the equation:

$(x + 6)^2 - 36 + (y - 3)^2 - 9 = 7.$

3. Simplify the equation

Combine like terms:

$(x + 6)^2 + (y - 3)^2 - 45 = 7.$

Add 45 to both sides to get the equation in standard form:

$(x + 6)^2 + (y - 3)^2 = 52.$

4. Identify the center and radius

The equation $(x + 6)^2 + (y - 3)^2 = 52$ is now in the standard form of a circle. From this, we can see:

• Center: $(h, k) = (-6, 3)$
• Radius: $r = \sqrt{52} = 2\sqrt{13}$

Conclusion

The center of the circle is at $(-6, 3)$ and the radius is $2\sqrt{13}$.

Would you like further clarification on the process of completing the square or have any other questions?

Further Related Questions

1. How do you find the equation of a circle given the endpoints of its diameter?
2. What is the process for converting a general quadratic equation into the standard form of a circle?
3. How do you determine if a point lies inside, on, or outside a given circle?
4. What is the significance of the radius in relation to a circle's curvature?
5. How do changes in the coefficients of $x$ and $y$ affect the position and size of a circle?

Tip:

When dealing with conic sections, completing the square is a powerful technique to transform a general equation into a standard form, making it easier to identify key characteristics such as center, radius, foci, vertices, and asymptotes.