9.2. Ellipses Definition of Ellipse CHAPTER 9: CONIC SECTIONS 9.2. Ellipses Definition of Ellipse An ellipse is a locus of all points (x,y) such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant. P F1 F2 F1P + F2P = 2a

Parts of an ellipse Center (h,k) Major Axes: Minor Axes: Vertices: Co-vertices: Foci: Co-vertex Center (h,k) focus Major axis Vertex Vertex Minor axis Co-vertex There are TWO cases of an ellipse: Horizontal major axes and Vertical major axes

Properties and Equation of an ellipse Horizontal Major Axis and C(0,0): a2 > b2 a2 – b2 = c2 F1(–c, 0) F2 (c, 0) y x V1(–a, 0) V2 (a, 0) (0, b) (0, –b) O x2 a2 y2 b2 + = 1 major axis = 2a minor axis = 2b

Rules and Properties Horizontal Major Axis and C(h,k): (x – h)2 a2 (y – k)2 b2 + = 1 a2 > b2 a2 – b2 = c2

Properties and Equation of an ellipse Vertical Major Axis and C(0,0): F1(0, –c) F2 (0, c) y x V1(0, –a) V2 (0, a) (b, 0) (–b, 0) O a2 > b2 a2 – b2 = c2 x2 b2 y2 a2 + = 1 major axis = 2a minor axis = 2b

Rules and Properties Vertical Major Axis and C(h,k): (x – h)2 b2 (y – k)2 a2 + = 1 a2 > b2 a2 – b2 = c2

ECCENTRICITY OF AN ELLIPSE The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth. In the 17th century, Johannes Kepler discovered that each planet travels around the sun in an elliptical orbit

ECCENTRICITY OF AN ELLIPSE One of the reasons it was difficult to detect that orbits are elliptical is that the foci of the planetary orbits are relatively close to the center, making the ellipse nearly circular. To measure the ovalness of an ellipse, we use the concept of eccentricity. DEFINITION: The eccentricity e of an ellipse is given by the ratio e = c/a e e1

PRACTICE EX. 1: Write equations of ellipses graphed in the coordinate plane

PRACTICE EX. 2: Sketch the graph of each ellipse. Identify the center, the vertices, the co-vertices, and the foci for each ellipse.

PRACTICE (x – 2)2 16 (y – 1)2 9 + = 1 center: (2, 1) EX.3: Find the coordinates of the center and vertices of an ellipse. Graph the ellipse. (x – 2)2 16 (y – 1)2 9 + = 1 center: (2, 1) vertices: (–2, 1), (6, 1)

PRACTICE (x – 2)2 16 (y – 1)2 9 + = 1 co-vertices: (2, 4), (2, –2) EX. 4: Find the coordinates of the co-vertices, and foci of an ellipse. Graph the ellipse. (x – 2)2 16 (y – 1)2 9 + = 1 co-vertices: (2, 4), (2, –2) foci: (2 – 7 , 1), (2 + 7 , 1)

PRACTICE EX.5: Graph the ellipse. 9x2 + 16y2 – 36x – 32y – 92 = 0 standard form: (x – 2)2 16 (y – 1)2 9 + = 1