Circle: Tangent to a line: Find the equation of the circle centered at (4, 11) and tangent to the line x + 2y + 4 = 0.
We can use the fact that at the tangent point, the radius is perpendicular to the tangent line. We can use the slope of the tangent line to get the slope of the line containing the radius. Then we can find the coordinates of the intersection of this line with the tangent line to get the coordinates of the point of tangency.

First, obtain the slope of the tangent line by writing its equation in the slope-intercept form:

becomes , or the slope of the tangent line is -1/2.

From the relationship of the slopes of perpendicular lines, we have

. This is the slope of the radius from the point of tangency to the center of the circle. We now find the equation of this line using the slope and the coordinates of the center in the point-slope form:

, or y - 11 = 2x - 8, or y = 2x + 3.

This line (a radius) and the tangent line intersect. Call the coordinates of the intersection . These coordinates must satisfy the equations of both lines:

. Solving for , we get

, or . We can substitute this result into the original equation to get

.

Next, we can find the radius of the circle by finding the distance from to the center of the circle. Using the definition of distance, we get

Finally, we can substitute in the equation of the circle using the coordinates of the center as h and k: h = 4, k = 11:

We can check this equation by substituting the coordinates of the point of intersection: (-2, -1):

, and the equation is correct.