To find the standard form of the equation of the circle with center $(4, -5)$ and tangent to the y-axis, we need to follow these steps:

Step 1: Recall the general equation of a circle

The standard form of the equation of a circle is given by: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center of the circle, and $r$ is its radius.

Step 2: Determine the radius of the circle

The problem states that the circle is tangent to the y-axis. The distance between the center of the circle and the y-axis is the radius $r$. Since the center is at $(4, -5)$, the distance from the center to the y-axis is the horizontal distance, which is simply the x-coordinate of the center, $4$.

Thus, the radius of the circle is: $r = 4$

Step 3: Write the equation of the circle

Now, substitute the center $(h, k) = (4, -5)$ and the radius $r = 4$ into the standard form of the equation of a circle: $(x - 4)^2 + (y + 5)^2 = 4^2$ Simplifying: $(x - 4)^2 + (y + 5)^2 = 16$

Final Answer:

The standard form of the equation of the circle is: $(x - 4)^2 + (y + 5)^2 = 16$

Would you like more details or have any questions about this solution? Here are some related questions to extend your understanding:

1. How do you find the equation of a circle given a point on the circle and its center?
2. How do you determine the equation of a line tangent to a circle at a given point?
3. What is the geometric significance of the radius in the context of tangency to a line?
4. How can you find the points where a line intersects a circle?
5. What are the different forms of a circle's equation, and how are they derived?

Tip: The distance between the center of a circle and a tangent line is always equal to the radius.