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10-3 Ellipses Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

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**Warm Up If c2 = a2 – b2, find c if 1. a = 13, b = 5 c = ±12**

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**Objectives Write the standard equation for an ellipse.**

Graph an ellipse, and identify its center, vertices, co-vertices, and foci.

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**Vocabulary ellipse focus of an ellipse major axis**

vertices of an ellipse minor axis co-vertices of an ellipse

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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.

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**Example 1: Using the Distance Formula to Find the Constant Sum of an Ellipse**

Find the constant sum for an ellipse with foci F1 (3, 0) and F2 (24, 0) and the point on the ellipse (9, 8). d = PF1 + PF2 Definition of the constant sum of an ellipse Distance Formula Substitute. Simplify. d = 27 The constant sum is 27.

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Check It Out! Example 1 Find the constant sum for an ellipse with foci F1 (0, –8) and F2 (0, 8) and the point on the ellipse (0, 10). d = PF1 + PF2 Definition of the constant sum of an ellipse Distance Formula Substitute. Simplify. d = 20 The constant sum is 20.

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**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.

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

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**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.

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**Example 2A: 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) Step 1 Choose the appropriate form of equation. x2 a2 = 1 y2 b2 The vertex is on the x-axis.

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**Step 2 Identify the values of a and b.**

Example 2A Continued Step 2 Identify the values of a and b. a = 6 The vertex (6, 0) gives the value of a. b = 4 The co-vertex (0, 4) gives the value of b. Step 3 Write the equation. x2 36 = 1 y2 16 Substitute the values into the equation of an ellipse.

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**Example 2B: 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 b2 The vertex is on the y-axis. Step 2 Identify the values of b and c. b = 5 The co-vertex (5, 0) gives the value of b. c = 3 The focus (0, 3) gives the value of c.

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**Step 3 Use the relationship c2 = a2 – b2 to find a2.**

Example 2B Continued Step 3 Use the relationship c2 = a2 – b2 to find a2. 32 = a2 – 52 Substitute 3 for c and 5 for b. a2 = 34 Step 4 Write the equation. y2 34 = 1 x2 25 Substitute the values into the equation of an ellipse.

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**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.

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**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.

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**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.

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**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

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

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**Example 3: Graphing Ellipses**

Graph the ellipse Step 1 Rewrite the equation as Step 2 Identify the values of h, k, a, and b. h = –4 and k = 3, so the center is (–4, 3). a = 7 and b = 4; Because 7 > 4, the major axis is horizontal.

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Example 3 Continued 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).

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Check It Out! Example 3a Graph the ellipse Step 1 Rewrite the equation as Step 2 Identify the values of h, k, a, and b. h = 0 and k = 0, so the center is (0, 0). a = 8 and b = 5; Because 8 > 5, the major axis is horizontal.

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**Check It Out! Example 3a Continued**

Graph the ellipse Step 3 The vertices are (±8, 0) or (8, 0) and (–8, 0), and the co-vertices are (0, ±5,), or (0, 5) and (0, –5).

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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.

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**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).

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**Example 4: Engineering Application**

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. x2 50 = 1 y2 20

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Example 4 Continued Find the dimensions of the new park. Step 1 Find the dimensions of the original park. Because 50 > 20, the major axis of the park is horizontal. a2 = 50, so and the length of the park is b2 = 20, so and the width of the park is

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Example 4 Continued Step 2 Find the dimensions of the new park. The length of the park is The width of the park is

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Example 4 Continued B. Write an equation for the design of the new park. Step 1 Use the dimensions of the new park to find the values of a and b. For the new park, Step 2 Write the equation. The equation in standard form for the new park will be .

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Check It Out! Example 4 Engineers have designed a tunnel with the equation measured in feet. A design for a larger tunnel needs to be twice as wide and 3 times as tall. x2 64 = 1, y2 36

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**Check It Out! Example 4 Continued**

a. Find the dimensions for the larger tunnel. Step 1 Find the dimensions of the original tunnel. Because 64 > 36, the major axis of the tunnel is horizontal. a2 = 64, so a = 8 and the width of the tunnel is 2a = 16 ft. b2 = 36, so b = 6 and the height of the tunnel is 6 ft. Step 2 Find the dimensions of the larger tunnel. The width of the larger tunnel is 2(16) = 32 ft. The height of the larger tunnel is 3(6) = 18 ft.

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**Check It Out! Example 4 Continued**

b. Write an equation for the larger tunnel. Step 1 Use the dimensions of the larger tunnel to find the values of a and b. For the larger tunnel, a = 16 and b = 18. Step 2 Write the equation. The equation in standard form for the larger tunnel is x2 162 = 1 or = 1. y2 182 256 324

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Lesson Quiz: Part I 1. Find the constant sum for an ellipse with foci F1 (2, 0), F2(–6, 0) and the point on the ellipse (2, 6). 16 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).

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Lesson Quiz: Part II 3. Graph the ellipse

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