To find the standard form of the equation for a circle, we use the formula:

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

Where:

• $(h, k)$ is the center of the circle
• $r$ is the radius

Given:

• Center $(h, k) = (2, -11)$
• The circle passes through the point $(-2, -5)$

Step 1: Find the radius $r$

We can calculate the radius by finding the distance between the center $(2, -11)$ and the point on the circle $(-2, -5)$ using the distance formula:

$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the points $(x_1, y_1) = (2, -11)$ and $(x_2, y_2) = (-2, -5)$:

$r = \sqrt{((-2) - 2)^2 + ((-5) - (-11))^2}$ $r = \sqrt{(-4)^2 + (6)^2}$ $r = \sqrt{16 + 36} = \sqrt{52}$ $r = 2\sqrt{13}$

Step 2: Write the standard form of the equation

Now that we know the radius $r = 2\sqrt{13}$, we can substitute the values into the equation:

$(x - 2)^2 + (y + 11)^2 = (2\sqrt{13})^2$ $(x - 2)^2 + (y + 11)^2 = 52$

Thus, the standard form of the equation of the circle is:

$(x - 2)^2 + (y + 11)^2 = 52$

Would you like more details or have any other questions?

Here are 5 related questions to expand on this:

1. How do you convert the equation of a circle from standard form to general form?
2. How would the equation change if the radius were doubled?
3. What are some other methods for finding the radius of a circle?
4. Can you determine the diameter of the circle from this equation?
5. How do you graph a circle given its standard form?

Tip: When dealing with equations of circles, always ensure the radius is squared on the right side of the equation.