The standard form of the equation of a circle is:

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

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

Given:

• Center: $(0, -2)$
• A point on the circle: $(-4, -5)$

Step 1: Find the radius

The radius $r$ is the distance from the center of the circle to any point on the circle. We can use the distance formula:

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

Substituting the coordinates of the center $(0, -2)$ and the point $(-4, -5)$:

$r = \sqrt{(0 - (-4))^2 + (-2 - (-5))^2}$ $r = \sqrt{(4)^2 + (3)^2}$ $r = \sqrt{16 + 9} = \sqrt{25} = 5$

Step 2: Write the equation of the circle

Now that we know the radius is $5$, we can plug the values into the standard form:

$(x - 0)^2 + (y - (-2))^2 = 5^2$ $x^2 + (y + 2)^2 = 25$

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

$x^2 + (y + 2)^2 = 25$

Would you like further details or explanations?

Here are 5 related questions to expand on this:

1. How do you find the equation of a circle if given two points on the circle?
2. Can the equation of a circle be written in general form?
3. How do you determine if a point lies inside, on, or outside a circle?
4. What happens to the equation if the circle is centered at the origin?
5. How do you find the tangent to a circle at a given point?

Tip: The radius is always a positive value, representing the distance from the center to any point on the circle.