Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation

Lesson 10-3 Ellipses Objective: To use and determine the standard and general forms of the equation of an ellipse To graph ellipses

Ellipse - Definition ellipse foci An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d 1 + d 2 = a constant value. center The center of the ellipse is the midpoint of the line segment that joins the foci

major axisminor axis. An ellipse has 2 axes of symmetry. The longer one contains the foci and is the major axis. The shorter one is called the minor axis. F 1 : (-c,0)F 2 : (c,0) Minor Axis Major Axis Vertex

Ellipse - Equation The equation of an ellipse centered at (0, 0) is …. where c 2 = a 2 – b 2 and c is the distance from the center to the foci. Shifting the graph over h units and up k units, the center is at (h, k) and the equation is where c 2 = a 2 – b 2 and c is the distance from the center to the foci.

Equation of an Ellipse: Center at (0,0); Foci at (0, c) and (0, -c); Major Axis is Vertical where a 2 ≥ b 2 and b 2 = a 2 - c 2 The major axis is the y - axis. The vertices are at (0, -a) and (0, a)

aa b b c c Ellipse - Graphing where c 2 = a 2 – b 2 and c is the distance from the center to the foci. Vertices are “a” units on the major axis and “b” units on the minor axis. The foci are “c” units in the direction of the major axis.

Ellipse – Table Center:(h, k) Vertices: Foci:c 2 = a 2 – b 2 a 2 ≥ b 2 If “a” is under “y”, the major axis is vertical If “a” is under “x”, the major axis is horizontal

Ellipse - Graphing Graph: Center:(2, -3) Distance to vertices in x direction: 4 Distance to vertices in y direction: 5 Distance to foci: c 2 =25-16 c 2 = 9 c = 3

Ellipse - Graphing Graph: Complete the squares.

Ellipse - Graphing Graph: Center:(-1, 3) Distance to vertices in x direction: Distance to vertices in y direction: Distance to foci: c 2 =| | c 2 = 15 c = 5

Ellipse – Find An Equation Find an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4. The center is the midpoint of the foci or (2, -3). The minor axis has a length of 4 and the vertices must be 2 units from the center. Start writing the equation.

Ellipse – Find An Equation c 2 = |a 2 – b 2 |. Since the major axis is in the x direction, a 2 > 16 9 = a 2 – 16 a 2 = 25 Replace a 2 in the equation.

Ellipse – Find An Equation The equation is:

Practice Find the coordinates of the center, foci and vertices of the ellipse. Then graph