example 2 write an equation given a vertex and a co-vertex write an equation of the ellipse that has...
Post on 17-Jan-2016
221 Views
Preview:
TRANSCRIPT
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.
top related