Objectives and Vocabulary Write the standard equation for an ellipse. Graph an ellipse, and identify its center, vertices, co-vertices, and foci. ellipse focus of an ellipse major axis minor axis vertices and co-vertices of an ellipse

Notes 1. Graph the ellipse . 2. Write an equation in standard form for each ellipse with the center at the origin. A. Vertex at (0, 5); co-vertex at (1, 0). B. Vertex at (5, 0); focus at (–2, 0). 3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1) 4. A city park in the form of an ellipse is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park. 2

Example 1 : Graphing Ellipses Graph the ellipse The vertices are (±8, 0) or (8, 0) and (–8, 0), and the co-vertices are (0, ±5,), or (0, 5) and (0, –5).

The standard form of an ellipse centered at (0, 0) depends on whether the major axis is horizontal or vertical.

If you pulled the center of a circle apart into two points, it would stretch the circle into an ellipse. An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F1 and F2, called the foci (singular: focus), is the constant sum d = PF1 + PF2. This distance d can be represented by the length of a piece of string connecting two pushpins located at the foci. You can use the distance formula to find the constant sum of an ellipse.

Example 2: Graphing Ellipses Graph the ellipse Step 3 The vertices are (–4 ± 7, 3) or (3, 3) and (–11, 3), and the co-vertices are (–4, 3 ± 4), or (–4, 7) and (–4, –1).

Example 3A: Using Standard Form to Write an Equation for an Ellipse Write an equation in standard form for each ellipse with center (0, 0). Vertex at (6, 0); co-vertex at (0, 4) Identify major axis - The vertex is on the x-axis. Step 1 x2 a2 + = 1 y2 b2 x2 36 + = 1 y2 16 Substitute the values into the equation of an ellipse. Step 2

Example 3B: Using Standard Form to Write an Equation for an Ellipse Write an equation in standard form for each ellipse with center (0, 0). Co-vertex at (5, 0); focus at (0, 3) Step 1 Choose the appropriate form of equation. y2 a2 + = 1 x2 52 The vertex is on the y-axis. Step 2 Use the foci to complete equation y2 34 + = 1 x2 25

Notes 1. Graph the ellipse . 2. Write an equation in standard form for each ellipse with the center at the origin. A. Vertex at (0, 5); co-vertex at (1, 0). B. Vertex at (5, 0); focus at (–2, 0). 3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1) 4. A city park in the form of an ellipse is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park. 9

A. Vertex at (0, 5); co-vertex at (1, 0). Notes 1. Graph the ellipse . 2. Write an equation in standard form for each ellipse with the center at the origin. A. Vertex at (0, 5); co-vertex at (1, 0). B. Vertex at (5, 0); focus at (–2, 0). 3. Write an equation of a shifted ellipse with the vertices (2, 5) and (2, -3) and co-Vertices of (4, 1) and (0, 1)

Notes #4: Engineering Application 4. A city park in the form of an ellipse with equation , measured in meters, is being renovated. The new park will have a length and width double that of the original park. Write the equation for the new park. x2 49 + = 1 y2 25

Step 1: Find the dimensions of the existing park. Length of major axis = 14 Length of minor axis = 10 x2 49 + = 1 y2 25 Step 2: Find the dimensions of the new park. Length of major axis = 28 Length of minor axis = 20 Step 3: Write the equation for the new park. x2 198 + = 1 y2 100

The values a, b, and c are related by the equation c2 = a2 – b2 The values a, b, and c are related by the equation c2 = a2 – b2. Also note that the length of the major axis is 2a, the length of the minor axis is 2b, and a > b.

Ellipses may also be translated so that the center is not the origin.

Ellipses: Extra Info The following power-point slides contain extra examples and information. Reminder: Lesson Objectives Write the standard equation for an ellipse. Graph an ellipse, and identify its center, vertices, co-vertices, and foci. 15

Instead of a single radius, an ellipse has two axes Instead of a single radius, an ellipse has two axes. The longer the axis of an ellipse is the major axis and passes through both foci. The endpoints of the major axis are the vertices of the ellipse. The shorter axis of an ellipse is the minor axis. The endpoints of the minor axis are the co-vertices of the ellipse. The major axis and minor axis are perpendicular and intersect at the center of the ellipse.

Notes #3: Solutions 3. Graph the ellipse .

Vertex at (9, 0); co-vertex at (0, 5) Check It Out! Example 2a Write an equation in standard form for each ellipse with center (0, 0). Vertex at (9, 0); co-vertex at (0, 5) Step 1 Choose the appropriate form of equation. x2 a2 + = 1 y2 b2 The vertex is on the x-axis.

Check It Out! Example 2a Continued Step 2 Identify the values of a and b. a = 9 The vertex (9, 0) gives the value of a. b = 5 The co-vertex (0, 5) gives the value of b. Step 3 Write the equation. x2 81 + = 1 y2 25 Substitute the values into the equation of an ellipse.

Co-vertex at (4, 0); focus at (0, 3) Check It Out! Example 2b Write an equation in standard form for each ellipse with center (0, 0). Co-vertex at (4, 0); focus at (0, 3) Step 1 Choose the appropriate form of equation. y2 a2 + = 1 x2 b2 The vertex is on the y-axis.

Check It Out! Example 2b Continued Step 2 Identify the values of b and c. b = 4 The co-vertex (4, 0) gives the value of b. c = 3 The focus (0, 3) gives the value of c. Step 3 Use the relationship c2 = a2 – b2 to find a2. 32 = a2 – 42 Substitute 3 for c and 4 for b. a2 = 25 Step 4 Write the equation. y2 25 + = 1 x2 16

Check It Out! Example 3b Graph the ellipse. Step 1 Rewrite the equation as Step 2 Identify the values of h, k, a, and b. h = 2 and k = 4, so the center is (2, 4). a = 5 and b = 3; Because 5 > 3, the major axis is horizontal.

Check It Out! Example 3b Continued Graph the ellipse. (2, –1) (7, 4) Step 3 The vertices are (2 ± 5, 4) or (7, 4) and (–3, 4), and the co-vertices are (2, 4 ± 3), or (2, 7) and (2, 1).