The eccentricity of an ellipse always satisfies 0 0, b > 0 y 2 a 2 x2 = 1, a > 0, b > 0 b2 Vertices: (±a, 0) (0, ±a) Transverse Axis: Horizontal, length 2a Vertical, length 2a Foci: (±c, 0) (0, ±c) c 2 = a 2 + b 2 c 2 = a 2 + b 2 Asymptotes: y = ± b a x y = ± a b x The way you can tell the difference between these two types is that the transverse axis corresponds to the variable which is positive in the equation (doesn t have the negative sign in front of it). The eccentricity e of an ellipse is a measure of how elongated the ellipse is. (The x-variable corresponds with a horizontal major axis and the y-variable corresponds with a vertical major axis.) Once you know which is the major axis, you know that the vertices and foci must lie on this axis. Parabola with Vertical Axis Parabola with Horizontal Axis Equation: x 2 = 4py y 2 = 4px Vertex: (0, 0) (0, 0) Focus: (0, p) (p, 0) Directrix: y = p x = p Focal diameter: 4p 4p Opens: upward if p > 0 to the right if p > 0 downward if p b > 0 x 2 b 2 + y2 a 2 = 1, a > b > 0 Vertices: (±a, 0) (0, ±a) Major Axis: Horizontal, length 2a Vertical, length 2a Minor Axis: Vertical, length 2b Horizontal, length 2b Foci: (±c, 0) (0, ±c) c 2 = a 2 b 2 c 2 = a 2 b 2 The way to distinguish between these two types is that the larger number a always appears in the denominator below the variable that corresponds with the major axis. The focal diameter of a parabola is the length of the line segment that runs through the focus perpendicular to the axis with endpoints on the parabola. The axis of symmetry is the line that runs throught the focus perpendicular to the directrix. The vertex V of a parabola lies halfway between the focus and the directrix.
A parabola is the set of points in the plane equidistant from a fixed point F called the focus and a fixed line l called the directrix. All parabolas in this section have their vertex at the origin. In this section, parabolas are discussed from a geometrical viewpoint. 1 10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions.
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