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 x2

32 +y2

42 = 1 x2 9+

y2

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 =400

2= 200

and b = 200 2

= 100

= 1An equation is = 1or x2

2002 + y2

1002

x2

40,000 +

y2

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 c2 = a2 – b2.

SOLUTION

42 = 82 – b2

b2 = 82 – 42 = 48

EXAMPLE 4 Write an equation given a vertex and a focus

b = 48, or 34

ANSWER

An equation is x2

82 + = 1 or x2

64+

y2

48 = 1 y2

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 x2

72 +y2

22 = 1 orx2

49+

y2

4= 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 x2

(– 5)2

+y2

62 = 1 orx2

25+

y2

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 c2 = b2 – a2.

(– 3)2 = 82 – a2

a2 = 82 – (– 3)2

a = 55

GUIDED PRACTICE for Examples 2, 3 and 4

or x2

55+

y2

64 = 1

ANSWER

An equation isy2

82+ = 1 x2

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 c2 = a2 – b2.

32 = (– 5)2 – b2

b2 = 25 – 9

a = 16 = + 4

GUIDED PRACTICE for Examples 2, 3 and 4

ANSWER

An equation is x2

52 +y2

42 = 1 orx2

25+

y2

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

= 1An equation is = 1or x2

1252 + y2

1752

x2

15625 +

y2

30625

The major axis is horizontal, with b =and a = 250

2= 125

350 2

= 175

GUIDED PRACTICE for Examples 2, 3 and 4

STEP 2

The area is A = π(125) (175) 68,700 square meters.