Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)

10.4 Ellipses What are the major parts of an ellipse? What are the standard forms of an ellipse equation? How do you find the value of the foci c when given a & b? How do you graph an ellipse? How do you find the equation of an ellipse from a graph?

Definition of Ellipse An ellipse is the set of all points P in a plane 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

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

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

Example 1 Write the standard equation for an ellipse with foci at (-8,0) and (8,0) and with a major axis of 20. Sketch the graph. -8 -6 -4 -2 2 4 6 8 length of major axis: 2a 2a = 20, so a = 10 a2 – b2 = c2 102 – b2 = 82 b2 = 100 - 64 b2 = 36, so b = 6 x2 100 y2 36 + = 1

Example 2 Find the vertices and co-vertices of the ellipse. x2 16 y2 49 + = 1 vertices: (0,7) and (0,-7) co-vertices: (4,0) and (-4,0)

Example 3 Write the standard equation of the ellipse. x2 16 y2 64 + length of major axis: 2a 2a = 16, so a = 8 -8 -6 -4 -2 2 4 6 8 length of minor axis: 2b 2b = 8, so b = 4 x2 16 y2 64 + = 1

Practice Write the standard equation for an ellipse with foci at (5,0) and (-5,0) and with vertices at (9,0) and (-9,0). Sketch the graph.

What are the major parts of an ellipse? Vertices, co-vertices, foci, center, major & minor axis What are the standard forms of an ellipse equation? How do you find the value of the foci c when given a & b? c2= a2 – b2 How do you graph an ellipse? Plot the vertices, the co-vertices and draw the ellipse. How do you find the equation of an ellipse from a graph? a = vertex of major axis goes into the denominator under major axis letter, and b = co-vertex of minor axis goes into the denominator under the minor axis letter in the formula.

Assignment 10.4 p.612, 19-67 Skip 41, 47

Warm-Up Write the standard equation for an ellipse with foci at (-5,0) and (5,0) and with a major axis of 18. Sketch the graph.

10.4 Day 2 Ellipses What is the standard equation for an ellipse if the vertex has been translated?

Standard Equation of a Translated Ellipse Horizontal Major Axis: (x – h)2 a2 (y – k)2 b2 + = 1 a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

Standard Equation of a Translated Ellipse Vertical Major Axis: (x – h)2 b2 (y – k)2 a2 + = 1 a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

Example 1 An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. 4x2 – 16x + 9y2 + 18y = 11 4(x2 – 4x) + 9(y2 + 2y) = 11 4(x2 – 4x + 4) + 9(y2 + 2y + 1) = 11 + 4(4) + 9(1) 4(x – 2)2 + 9(y + 1)2 = 36

Example 1 Write an equation of the ellipse with foci at (3,5) and (3,−1) and vertices at (3,6) and (3,−2). Plot the given points and make a rough sketch. The ellipse has a vertical axis, so its equation is in the form of: -6 -4 -2 2 4 6 Find the center which is halfway between the vertices. Find a and c a = 6−2=4 c = 5−2=3 Find b using b2= a2−c2

What is the standard equation for an ellipse if the vertex has been translated?

Assignment 10.4 day 2 p. 612, 24-60 even p. 628, 17, 18, 24, 25