Ellipses quick check answers
1) A parabola is the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
2) An ellipse is the set of all points in a plane, such that the sum of the distances from two fixed points (the foci) is constant for every point on the curve.
3) The distance between the foci of an ellipse is given by the formula 2a, where a is the length of the semi-major axis.
4) The distance between the vertices of an ellipse is given by the formula 2b, where b is the length of the semi-minor axis.
5) The standard form equation of an ellipse with its center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b represent the lengths of the semi-major and semi-minor axes, respectively.
6) The eccentricity of an ellipse is a measure of its elongation, and is given by the formula c/a, where c is the distance from the center of the ellipse to each focus, and a is the length of the semi-major axis.
7) An ellipse becomes a circle when the lengths of the semi-major and semi-minor axes are equal.
8) The equation of a horizontal ellipse with its center at (h, k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
9) The equation of a vertical ellipse with its center at (h, k) is (x-h)^2/b^2 + (y-k)^2/a^2 = 1.
10) Ellipses have both a major axis (the longest diameter) and a minor axis (the shortest diameter), and the center of the ellipse is the midpoint of these axes.