To solve this problem, we need to use the standard form of the equation of a circle, which is given by:

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

Here:

• $(h, k)$ represents the coordinates of the center of the circle.
• $r$ represents the radius of the circle.

Given:

• The center of the circle is $(1, -8)$, so $h = 1$ and $k = -8$.
• The radius $r$ is $\sqrt{10}$.

Now, let's substitute these values into the equation:

$(x - 1)^2 + \left(y - (-8)\right)^2 = (\sqrt{10})^2$

Simplifying this equation:

$(x - 1)^2 + (y + 8)^2 = 10$

So, the correct equation of the circle is:

$(x - 1)^2 + (y + 8)^2 = 10$

This corresponds to the second option in the image.

Would you like more details or have any questions?

Here are five related questions to expand on this concept:

1. How would the equation change if the circle's center was at the origin?
2. What is the significance of the coefficients of $x$ and $y$ in the circle's equation?
3. How would the equation of the circle change if the radius was doubled?
4. Can you derive the equation of a circle given a different center and radius?
5. How would you determine the center and radius of a circle from a given equation in general form?

Tip: Always ensure that the terms are correctly expanded and simplified when dealing with the equation of a circle to avoid errors in identifying the correct form.