The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangent. Find the standard form of the equation of the circle with the center (4,-5) and tangent to the y-axis.

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.

Explore how to find the equation of a circle with center (4, -5) and tangent to the y-axis. This problem involves key geometric concepts like circles, tangents, and the equation of a circle. Detailed step-by-step solution provided. Suitable for students in Grades 9-11.

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