We are asked to find the equation of a circle with a center at $(3, -2)$ that is tangent to the y-axis. The general form of a circle's equation is:

$(x - A)^2 + (y - B)^2 = C$

where $(A, B)$ is the center of the circle, and $C$ is the square of the radius.

Step 1: Identify the center and radius.

• The center of the circle is given as $(3, -2)$, so $A = 3$ and $B = -2$.
• Since the circle is tangent to the y-axis, the distance from the center to the y-axis is the radius. The distance from $(3, -2)$ to the y-axis is 3 units (because the x-coordinate of the center is 3).

Thus, the radius $r = 3$.

Step 2: Write the equation of the circle.

The radius is squared in the equation of a circle, so:

$r^2 = 3^2 = 9$

Substituting $A = 3$, $B = -2$, and $r^2 = 9$ into the equation of the circle:

$(x - 3)^2 + (y + 2)^2 = 9$

This is the equation of the circle.

Final Answer:

$(x - 3)^2 + (y + 2)^2 = 9$

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. How is the distance formula used to find the radius in this problem?
2. What is the general equation of a circle and what do each of the terms represent?
3. How would the equation change if the circle was tangent to the x-axis instead?
4. What happens to the equation if the center is moved to a different location?
5. How would the equation change if the circle's radius doubled?

Tip: The distance from a point to the y-axis is simply the absolute value of the x-coordinate of the point!