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EXAMPLE 2

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.

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EXAMPLE 2

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  1. 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.

  2. + + 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

  3. EXAMPLE 3 Solve a multi-step problem Lightning When lightning strikes, an elliptical region where the strike most likely hit can often be identified. Suppose it is determined that there is a 50% chance that a lightning strike hit within the elliptical region shown in the diagram. • Write an equation of the ellipse. • The area Aof an ellipse is A = π ab. Find the area of the elliptical region.

  4. The major axis is horizontal, with a = = 200 400 2 200 2 andb = = 100 y2 1002 x2 40,000 y2 10,000 x2 2002 or = 1 = 1 + An equation is + The area isA = π(200)(100) 62,800 square meters. EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 STEP 2

  5. 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

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

  7. + + x2 72 x2 49 y2 22 y2 4 ANSWER = 1 = 1 An equation is or 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) SOLUTION Because the vertex is on the y - axis and the co-vertex is on the y - axis, the major axis is vertical with a = 7, and the minor axis is horizontal with b = 2.

  8. + y2 62 x2 25 y2 36 ANSWER x2 (– 5)2 = 1 = 1 An equation is or + for Examples 2, 3 and 4 GUIDED PRACTICE 5.Vertex: (0, 6); co-vertex: ( – 5, 0) SOLUTION Because the vertex is on the y - axis and the co-vertex is on the y - axis, the major axis is vertical with a = – 5, and the minor axis is horizontal with b = 6.

  9. a = 55 for Examples 2, 3 and 4 GUIDED PRACTICE 6.Vertex: (0, 8); focus: ( 0, – 3) SOLUTION Make a sketch of the ellipse. Because the given vertex and focus lie on the y - axis, the major axis is vertical, with b = 8 and c = –3. To find b, use the equation c2 = b2 – a2. (– 3)2 = 82 – a2 a2 = 82 – (– 3)2

  10. ANSWER y2 64 x2 x2 55 y2 82 + = 1 + = 1 An equation is or 55)2 ( for Examples 2, 3 and 4 GUIDED PRACTICE

  11. a = 16 for Examples 2, 3 and 4 GUIDED PRACTICE 7.Vertex: (– 5, 0); focus: ( 3, 0) SOLUTION Make a sketch of the ellipse. Because the given vertex and focus lie on the y - axis, the major axis is vertical, with a = 5 and c = 3. To find b, use the equation c2 = a2 – b2. 32 = (– 5)2 – b2 b2 = 25 – 9 = + 4

  12. + + x2 52 y2 42 x2 25 y2 16 ANSWER = 1 = 1 An equation is or for Examples 2, 3 and 4 GUIDED PRACTICE

  13. 250 2 350 2 y2 1752 x2 15625 y2 30625 x2 1252 + The major axis is horizontal, with b = = 175 anda = = 125 or = 1 = 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. SOLUTION STEP 1

  14. The area isA = π(125) (175) 68,700 square meters. for Examples 2, 3 and 4 GUIDED PRACTICE STEP 2

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