Give the standard equation of the circle with center (5, -6), tangent to the y-axis

To find the standard equation of the circle, we need the center and the radius. The center is given as \((5, -6)\).

Since the circle is tangent to the y-axis, the distance from the center to the y-axis is equal to the radius. The x-coordinate of the center is 5, so the radius \(r\) is 5.

The standard equation of a circle with center \((h, k)\) and radius \(r\) is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting \(h = 5\), \(k = -6\), and \(r = 5\), we get:

\[
(x - 5)^2 + (y + 6)^2 = 25
\]

So, the standard equation of the circle is:

\[
(x - 5)^2 + (y + 6)^2 = 25
\]

Would you like more details on how this is derived or any other related questions? 

Here are 8 follow-up questions:
1. How is the distance formula used in finding the radius?
2. What is the general form of a circle’s equation?
3. How do you find the equation of a circle given the diameter?
4. Can the equation of a circle be expressed in a different form?
5. How does the location of the tangent point affect the circle’s equation?
6. How do you find the equation of a circle tangent to both axes?
7. What happens to the equation if the circle is translated horizontally or vertically?
8. How does the radius affect the shape and size of the circle?

**Tip:** The standard form of a circle's equation is especially useful when identifying the circle's center and radius directly from the equation.

Explore the standard equation of a circle with center (5, -6), tangent to the y-axis. Learn how to derive the equation step-by-step, including the application of the circle's standard form formula. Suitable for Grades 10-12.

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