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.

EXAMPLE 2 Write an equation given a vertex and a co-vertex or ANSWER An equation is x232x232 + y242y242 = 1 x29x29 + y 2 16 = 1

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 A of an ellipse is A = π ab. Find the area of the elliptical region.

EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 The major axis is horizontal, with a = = 200 and b = = 100 = 1 An equation is = 1 or x y x 2 40,000 + y 2 10,000 STEP 2 The area is A = π(200)(100) 62,800 square meters.

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). Make a sketch of the ellipse. Because the given vertex and focus lie on the x - axis, the major axis is horizontal, with a = 8 and c = 4. To find b, use the equation c 2 = a 2 – b 2. SOLUTION 4 2 = 8 2 – b 2 b 2 = 8 2 – 4 2 = 48

EXAMPLE 4 Write an equation given a vertex and a focus b = 48, or 3 4 ANSWER An equation is x282x282 + = 1 or x y 2 48 = 1 y 2 3,) 2 (4

GUIDED PRACTICE for Examples 2, 3 and 4 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. ANSWER An equation is x272x272 + y222y222 = 1 or x y24y24 = 1

GUIDED PRACTICE for Examples 2, 3 and 4 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. ANSWER An equation is x 2 (– 5) 2 + y262y262 = 1 or x y 2 36 = 1

GUIDED PRACTICE for Examples 2, 3 and 4 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 c 2 = b 2 – a 2. (– 3) 2 = 8 2 – a 2 a 2 = 8 2 – (– 3) 2 a =55

GUIDED PRACTICE for Examples 2, 3 and 4 or x y 2 64 = 1 ANSWER An equation is y282y282 += 1 x 2 55 ) 2 (

GUIDED PRACTICE for Examples 2, 3 and 4 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 c 2 = a 2 – b = (– 5) 2 – b 2 b 2 = 25 – 9 a =16= + 4

GUIDED PRACTICE for Examples 2, 3 and 4 ANSWER An equation is x252x252 + y242y242 = 1 or x y 2 16 = 1

GUIDED PRACTICE for Examples 2, 3 and 4 8. What If ? In Example 3, suppose that the elliptical region is 250 meters from east to west and 350 meters from north to south. Write an equation of the elliptical boundary and find the area of the region. SOLUTION STEP 1 = 1 An equation is = 1 or x y x y The major axis is horizontal, with b = and a = = = 175

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