SECTION: 10-2 ELLIPSES WARM-UP Write the standard form of the equation of each parabola. 3. Find the equation of the tangent at point (2,–2) of the parabola x2= –2y.

ELLIPSE. An ellipse is the set of all points (x,y) the whose distances (d1,d2) from two distant fixed points (foci) is constant.

COMPONENTS OF AN ELLIPSE VERTICES. A line passing through both foci intersects the curve of the ellipse at the vertices. CENTER. The center of the ellipse is the midpoint between the vertices. MAJOR AXIS. The major axis it the chord with endpoints at the vertices and passing through the center of the ellipse and both foci. MINOR AXIS. The minor axis is the chord passing through the center of the ellipse and perpendicular to the major axis.

STANDARD EQUATION OF AN ELLIPSE STANDARD EQUATION OF AN ELLIPSE. The standard form of the equation of an ellipse, with center and major axis length of 2a and minor axis length of 2b, where 0<b<a is:

The foci lie on the major axis c units from the center of the ellipse with c2=a2–b2. The sum of the distances from any point on the ellipse to the two foci is constant. Using a vertex point, this constant sum is (a+c)+(a–c)=2a or simply the length of the major axis.

EXAMPLE 1. Write the standard form of the equation for each ellipse. a. Foci at (0,1) and (4,1) and major axis length 6. b. Foci at (2,1) and (2,5) and major axis length 8.

SKETCHING THE GRAPH OF AN ELLIPSE 1. Write the equation of the ellipse in standard form. 2. Determine and plot the center of the ellipse. 3. Determine the endpoints of the major axis by adding/subtracting the value of a to/from the coordinate of the center. 4. Determine the endpoints of the minor axis by adding/subtracting the value of b to/from the appropriate coordinate of the center.

EXAMPLE 2. Sketch the graph of the ellipse given by the equation

GENERAL EQUATION OF CONIC SECTIONS GENERAL EQUATION OF CONIC SECTIONS. The general equation of all conic sections is of the form TRANSCRIBE EQUATIONS 1. Group like terms. 2. Move the constant to the opposite side of the equal sign. 3. Complete the square on both the x- and y- variables. 4. Simplify the equation.

EXAMPLE 3. Write the standard form of each ellipse EXAMPLE 3. Write the standard form of each ellipse. Then state the coordinates of the center, vertices, and foci.

ECCENTRICITY. The eccentricity of a conic section describes the transformation, dilation or compression, of the conic section. The eccentricity of an ellipse is found using the formula:

EXAMPLE 4. Find the eccentricity of the following ellipses.

APPLICATION: ELLIPTICAL ORBIT

EXAMPLE 5. The moon travels about the earth in an elliptical orbit with the earth at one focus. The major and minor axes of the orbit have lengths of 768,806 kilometers and 767,746 kilometers respectively. Find the greatest and least distances from the earth’s center to the moon’s center.

CLASS WORK/HOMEWORK: SECTION: 10 – 2 PAGE: 710 – 711 PROBLEMS: 7, 9, 11, 29, 31, 39, 41, 47