The given equation is:

$x^2 + y^2 + 6x - 8y - 75 = 0$

We need to rewrite this equation in the standard form of a circle equation:

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

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

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

$x^2 + 6x + y^2 - 8y = 75$

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

• For $x^2 + 6x$, we take half of 6 (which is 3), square it (getting 9), and add 9 to both sides.
• For $y^2 - 8y$, we take half of -8 (which is -4), square it (getting 16), and add 16 to both sides.

So, we add 9 and 16 to both sides:

$x^2 + 6x + 9 + y^2 - 8y + 16 = 75 + 9 + 16$

$(x + 3)^2 + (y - 4)^2 = 100$

Step 3: Identify the radius

The equation is now in the standard form, where $r^2 = 100$, so:

$r = \sqrt{100} = 10$

Thus, the radius of the circle is $10$.

Answer: B) 10

Would you like further clarification or a breakdown of any part of the process?

Here are some related questions for you to explore:

1. How do you derive the standard form of the circle equation from a general quadratic equation?
2. How do you complete the square when the coefficient of $x^2$ or $y^2$ is not 1?
3. How can the center and radius be interpreted geometrically from the equation of a circle?
4. How can you find the equation of a circle when given its center and radius?
5. What is the general equation of a conic section, and how do different conic sections compare?

Tip: Always make sure to carefully complete the square when rewriting quadratic equations into their standard forms. This is key for easily identifying properties such as the center and radius of a circle.