Ellipses

The standard equation of a circle is (x - h)² + (y - k)² = r².

The center (h, k) can be found from the equation (x - h)² + (y - k)² = r², and the radius r is the square root of the constant on the right side.

The general form of the equation of an ellipse is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

In the equation (x - h)²/a² + (y - k)²/b² = 1, a is the length of the semi-major axis and b is the length of the semi-minor axis.

In an ellipse, the length of the semi-major axis (a) is always greater than or equal to the length of the semi-minor axis (b).

A conic section is a curve obtained by intersecting a cone with a plane. The types of conic sections include circles, ellipses, parabolas, and hyperbolas.

A circle is a special case of an ellipse where the lengths of the semi-major and semi-minor axes are equal (a = b).

The expression x² - a² can be factored using the difference of squares formula: x² - a² = (x - a)(x + a).

The center of an ellipse is the midpoint between its two foci and is the point from which the ellipse is symmetrically defined.

The coordinates of the foci of an ellipse can be found using the formula (h ± c, k) for horizontal ellipses and (h, k ± c) for vertical ellipses, where c = √(a² - b²).