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1. a = 9, b = 3

c = 3√10. b = 2√35. Solve a 2 + b 2 = c 2 for the missing value. 1. a = 9, b = 3. ANSWER. 2. a = 2, c = 12. ANSWER. 100. 25 y 2. x 2. 4 x 2. 100. 25. 100. 100. =. +. y 2 4. +. = 1. EXAMPLE 1. Graph an equation of an ellipse.

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1. a = 9, b = 3

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  1. c = 3√10 b = 2√35 Solve a2 + b2 = c2 for the missing value. 1.a = 9, b = 3 ANSWER 2.a = 2, c = 12 ANSWER

  2. 100 25y2 x2 4x2 100 25 100 100 = + y24 + = 1 EXAMPLE 1 Graph an equation of an ellipse Graph the equation 4x2 + 25y2 = 100. Identify the vertices, co-vertices, and foci of the ellipse. SOLUTION STEP 1 Rewrite the equation in standard form. 4x2 + 25y2 = 100 Write original equation. Divide each side by 100. Simplify.

  3. soc = 21 The foci are at( + 21 , 0),or about ( + 4.6, 0). EXAMPLE 1 Graph an equation of an ellipse STEP 2 Identify the vertices, co-vertices, and foci. Note that a2 = 25 and b2 = 4, so a = 5and b = 2. The denominator of the x2 - term is greater than that of the y2 - term, so the major axis is horizontal. The vertices of the ellipse are at (+a, 0) = (+5, 0).The co-vertices are at (0, +b) = (0, +2). Find the foci. c2= a2– b2= 52– 22= 21,

  4. EXAMPLE 1 Graph an equation of an ellipse STEP 3 Draw the ellipse that passes through each vertex and co-vertex.

  5. y2 x2 Graph the equation 1. Identify the + = 36 225 vertices, co-vertices, and foci of the ellipse. ANSWER vertices: (0, ± 15) co-vertices: (±6, 0) √21 ) foci: (0, + 3 EXAMPLE 2 Graph an equation of an ellipse Daily Homework Quiz

  6. + x2 16 The foci are at( + 7 , 0). y29 for Example 1 GUIDED PRACTICE Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. 1. = 1 ANSWER The vertices of the ellipse are at (+ 4, 0) and co-vertices are at (0, + 3).

  7. for Example 1 GUIDED PRACTICE

  8. y2 49 x2 + 36 The foci are at(0, ± 13 ). for Example 1 GUIDED PRACTICE Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. 2. = 1 ANSWER The vertices of the ellipse are at (0, + 7) and co-vertices are at (+ 6, 0).

  9. for Example 1 GUIDED PRACTICE

  10. for Example 1 GUIDED PRACTICE Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. 3. 25x2 + 9y2 = 225 ANSWER The vertices of the ellipse are at (0, + 5), and co-vertices are at (+ 3, 0). The foci are at(0, + 4).

  11. for Example 1 GUIDED PRACTICE

  12. EXAMPLE 2 Write an equation given a vertex and a co-vertex Write an equation of the ellipse that has a vertex at (0, 4), a co-vertex at (– 3, 0), and center at (0, 0). SOLUTION Sketch the ellipse as a check for your final equation. By symmetry, the ellipse must also have a vertex at (0, – 4) and a co-vertex at (3, 0). Because the vertex is on the y - axis and the co-vertex is on the x - axis, the major axis is vertical with a = 4, and the minor axis is horizontal with b = 3.

  13. + + x2 32 y2 42 x2 9 y2 16 = 1 EXAMPLE 2 Write an equation given a vertex and a co-vertex ANSWER = 1 An equation is or

  14. EXAMPLE 4 Write an equation given a vertex and a focus Write an equation of the ellipse that has a vertex at (– 8, 0), a focus at (4, 0), and center at (0, 0). SOLUTION Make a sketch of the ellipse. Because the given vertex and focus lie on the x - axis, the major axis is horizontal, with a = 8and c = 4. To find b, use the equation c2 = a2 – b2. 42 = 82 – b2 b2 = 82 – 42 = 48

  15. b = or 48, 3 4 + ANSWER y2 48 x2 64 y2 x2 82 = 1 = 1 An equation is or + (4 3) EXAMPLE 4 Write an equation given a vertex and a focus 2

  16. + x2 49 y2 4 An equation is = 1 for Examples 2, 3 and 4 GUIDED PRACTICE Write an equation of the ellipse with the given characteristics and center at (0, 0). 4.Vertex: (7, 0); co-vertex: (0, 2) ANSWER

  17. + x2 25 y2 36 = 1 An equation is for Examples 2, 3 and 4 GUIDED PRACTICE Write an equation of the ellipse with the given characteristics and center at (0, 0). 5.Vertex: (0, 6); co-vertex: ( – 5, 0) ANSWER

  18. y2 64 x2 55 An equation is + = 1 for Examples 2, 3 and 4 GUIDED PRACTICE Write an equation of the ellipse with the given characteristics and center at (0, 0). 6.Vertex: (0, 8); focus: ( 0, – 3) ANSWER

  19. + x2 25 y2 16 = 1 An equation is for Examples 2, 3 and 4 GUIDED PRACTICE Write an equation of the ellipse with the given characteristics and center at (0, 0). 7.Vertex: (– 5, 0); focus: ( 3, 0) ANSWER

  20. x2 15,625 y2 30,625 + = 1 An equation is for Examples 2, 3 and 4 GUIDED PRACTICE 8. What If ? In Example 3, suppose that the elliptical region is 250meters from east to west and 350meters from north to south. Write an equation of the elliptical boundary and find the area of the region. ANSWER The area is about 68,700 square meters.

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