Key Concepts:
- Distance between two Points: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
- Midpoint Formula: (M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})).
- The center of a circle is the midpoint of two endpoints of a diameter.
- Intersection with Axes: Whether a circle touches or crosses the x- or y-axis can be determined by the center’s coordinates and radius.
Skills Covered:
- Calculating distances between two points.
- Writing the circle's equation given two endpoints of a diameter.
- Identifying the number of intersections of a circle with x- or y-axis.
Example Problem:
- A circle has a diameter with two endpoints ((3, -2)) and ((-3, 6)). Write the equation of the circle.
- A circle has the equation ((x - 4)^2 + (y + 2)^2 = 25). How many intersections does it have with the y-axis?