Warm Up The latus rectum of a conic

Warm Up The latus rectum of a conic section is the chord through a focus parallel to the directrix.

Standard Form of a Parabola • The standard form of the equation of a parabola with a vertical axis and: • vertex at (h, k) • focus at (h, k + p) • directrix y = k - p (x – h)2 = 4 p(y – k), p 0. • The standard form of the equation of a parabola with a horizontal axis and: • vertex at (h, k) • focus at (h + p, k) • directrix at x = h - p (y – k)2 = 4 p(x – h), p 0

9. 2 Ellipses E 08 Pre Calc

TSW’s • TSW write equations of ellipses in standard form. [9. 2] • TSW use properties of ellipses to model and solve applied problems. [9. 2] • TSW find eccentricities of ellipses. [9. 2]

Ellipse • An ellipse is the set of all points (x, y) in a plane, the sum of whose distances from two distinct fixed points (foci) is constant. D 1+D 2 is a constant where D 1 is the distance from focus 1 to a fixed point on the ellipse and D 2 is the distance from focus 2 to that same point on the ellipse.

• Center/Origin-midpoint between the two foci • Major Axis -Goes through focus and starts and ends at the edge of the ellipse • Minor axis -chord perpendicular to major axis through center Covertex

Equation for Ellipses (c 2=a 2 -b 2) Center (h, k) Horizontal major axis: 2 a, Vertical minor axis: 2 b, Vertices at: (h a, k), Covertices at: (h, k b), Foci at (h c, 0) Centered: (h, k) Vertical major axis: 2 a, Horizontal Minor axis of length: 2 b, Vertices at (h, k a), Covertices at (h b, k), Foci at (h, k c)

Center (h, k)

Example 1. Find the center, vertices, and foci of the ellipse given by 9 x 2 + 4 y 2 = 36. Recall foci are c units from center on the major axis where c 2=a 2 -b 2.

Example 2. Find the standard form of the equation of the ellipse centered at the origin with major axis of length 10 and foci at ( 3, 0). Recall foci are c units from center on the major axis where c 2=a 2 -b 2.

Example 3. Sketch the graph of the following ellipse: 25 x 2 + 9 y 2 – 200 x + 36 y + 211 = 0 Recall foci are c units from center on the major axis where c 2=a 2 -b 2.

Application: Example 4. A passageway in a house is to have straight sides and a semielliptically-arched top. The straight sides are 5 feet tall, and the passageway is 7 feet tall at its center and 6 feet wide. Where are the foci to be located to make the template for the arch?

Eccentricity • The eccentricity of an ellipse is e = c/a, where 0 < e < 1. • The eccentricity is a measure of the “ovalness” of the ellipse. – If the eccentricity is close to 0, then the ellipse is close to being circular. – If the eccentricity is close to 1, then the ellipse is flatter.

• Find an equation of the ellipse with vertices (0, ± 8) and eccentricity e=1/2. (Recall e = c/a)