The problem asks to write the standard form of the equation of the circle shown in the diagram.

The standard form of a circle's equation is: $(x - h)^2 + (y - k)^2 = r^2$ where:

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

From the diagram, we can observe the following:

• The center is at $(-3, 3)$.
• The radius can be found by measuring the distance from the center $(-3, 3)$ to a point on the circle, like $(5, 3)$.

The distance between these two points is simply the horizontal distance (since the $y$-coordinates are the same): $r = |5 - (-3)| = 8$ Thus, the radius $r = 8$.

Now, we can write the equation of the circle using the standard form: $(x + 3)^2 + (y - 3)^2 = 8^2$ which simplifies to: $(x + 3)^2 + (y - 3)^2 = 64$

This is the standard form of the circle's equation.

Do you have any questions or need further details?

Here are 5 related questions to consider:

1. How do you find the center and radius of a circle from an equation?
2. What is the general formula for the distance between two points?
3. How would the equation change if the center were moved?
4. How can you determine the radius if it's not easily visible on the graph?
5. How do we convert the standard form of a circle into the general form?

Tip: The formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.