To solve this problem, we need to analyze the given equation of the circle:

$x^2 + y^2 - 8x + 10y - 12 = 0$

We will rewrite this equation in the standard form of a circle, which is:

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

where $(h, k)$ is the center of the circle and $r$ is the radius.

Step 1: Completing the Square

1. Group the terms involving $x$ and $y$: $(x^2 - 8x) + (y^2 + 10y) = 12$

2. Complete the square for $x$:

   • For $x^2 - 8x$, take half of $-8$, which is $-4$, and square it to get $16$.
   • Add $16$ to both sides.

   $(x^2 - 8x + 16) = 12 + 16$

3. Complete the square for $y$:

   • For $y^2 + 10y$, take half of $10$, which is $5$, and square it to get $25$.
   • Add $25$ to both sides.

   $(x^2 - 8x + 16) + (y^2 + 10y + 25) = 12 + 16 + 25$

4. Rewrite the equation:

   $(x - 4)^2 + (y + 5)^2 = 53$

Step 2: Identifying the Center and Radius

• The center $(h, k)$ of the circle is $(4, -5)$.
• The radius $r$ is $\sqrt{53}$.

Conclusion

• Statement I: The center of the circle is $(4, -5)$. This is true.
• Statement II: The radius of the circle is $\sqrt{53}$. This is also true.

Answer

The correct answer is: C. Both are true

Would you like further explanation on any part of this process?

Related Questions

1. How do you find the center and radius of a circle from its general equation?
2. What is the method of completing the square, and why is it useful for circle equations?
3. Can we use a different approach to verify the radius of a circle?
4. How do transformations affect the standard form of a circle equation?
5. What other types of conic sections can be derived by rearranging terms similarly?

Tip

When completing the square, always remember to balance the equation by adding the necessary constants to both sides.