To find the equation of the circle with a center at $(2, -1)$ and passing through the point $(3, 6)$, we'll follow these steps:

Step 1: Use the general equation of a circle

The general 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 $(h, k) = (2, -1)$

The equation becomes:

$(x - 2)^2 + (y + 1)^2 = r^2$

Step 2: Use the point $(3, 6)$ to find the radius

Substitute the point $(3, 6)$ into the equation to find the radius $r$:

$(3 - 2)^2 + (6 + 1)^2 = r^2$

Simplify the equation:

$1^2 + 7^2 = r^2$

$1 + 49 = r^2$

$r^2 = 50$

Step 3: Write the equation of the circle

Now, substitute $r^2 = 50$ back into the circle equation:

$(x - 2)^2 + (y + 1)^2 = 50$

This is the equation of the circle with center $(2, -1)$ and passing through the point $(3, 6)$.

Would you like further details or have any other questions? Here are some related topics you might find useful:

1. How to derive the general form of the circle's equation from the standard form.
2. Steps to find the radius if another point on the circle is known.
3. Converting the equation of a circle from standard to general form.
4. Understanding how the radius affects the size and position of the circle.
5. Analyzing how a circle's equation changes with different centers.

Tip: The distance formula is key in finding the radius when you know the center and a point on the circle.