Give the standard equation of the circle with center (-6, 7), tangent to the x-axis.

To find the standard equation of a circle with a given center and tangent to the x-axis, we can follow these steps:

### Step 1: Identify the center and radius
The center of the circle is given as \((-6, 7)\). Since the circle is tangent to the x-axis, the radius \(r\) of the circle is the distance from the center to the x-axis.

The distance from the center \((-6, 7)\) to the x-axis is simply the y-coordinate of the center, which is 7. Therefore, the radius \(r = 7\).

### Step 2: Write the standard equation of the circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is:

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

Substituting the values \(h = -6\), \(k = 7\), and \(r = 7\):

\[
(x + 6)^2 + (y - 7)^2 = 7^2
\]

Simplifying further:

\[
(x + 6)^2 + (y - 7)^2 = 49
\]

### Final Answer:
The standard equation of the circle is:

\[
(x + 6)^2 + (y - 7)^2 = 49
\]

Would you like more details on any part of the solution? 

Here are some related questions for further exploration:
1. What is the equation of a circle centered at (0, 0) and tangent to the y-axis?
2. How does the equation change if the circle is tangent to the y-axis instead?
3. Can you find the points where this circle intersects the y-axis?
4. How would the equation change if the radius were doubled?
5. What is the relationship between the radius and the distance from the center to the tangent line?
6. How do you find the tangent line to a circle at a given point?
7. Can you determine the equation of a circle given its center and a point on the circle?
8. How do you convert the equation of a circle to polar coordinates?

**Tip:** When a circle is tangent to a line, the radius of the circle is perpendicular to that line at the point of tangency. This property can simplify finding equations in certain problems.

Discover how to find the standard equation of a circle with center (-6, 7) that is tangent to the x-axis. Learn step-by-step with clear explanations and formulas used. Suitable for high school students and beyond.

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