Transcript Document

Warm-Up 1/09
1.
B
2.
G
Rigor:
You will learn how to identify, analyze and graph
equations of ellipses and circles, and how to write
equations of ellipses and circles.
Relevance:
You will be able to use graphs and equations of
ellipses and circles to solve real world problems.
7-2 Ellipses and
Circles
2b
2c
2a
Example 1: Graph the ellipse given by the equation.
𝑥−3
36
2
𝑦+1
+
9
2
=1
ℎ = 3 𝑘 = −1 𝑎 = 36 = 6 𝑏 = 9 = 3
𝑐 = 36 − 9 = 27 = 3 3
Orientation: horizontal
Center ℎ, 𝑘 : 3, −1
foci ℎ ± 𝑐, 𝑘 : 3 ± 3 3, −1
vertices ℎ ± 𝑎, 𝑘 : −3, −1 and 9, −1
co-vertices ℎ, 𝑘 ± 𝑏 : 3, −4 and 3, 2
major axis 𝑦 = 𝑘: 𝑦 = −1
minor axis 𝑥 = ℎ: 𝑥 = 3
•
•F•
•
•
• •F
Example 2a: Write an equation for an ellipse with given
characteristics.
major axis from (– 6, 2) to (– 6, – 8); minor axis from (– 3, – 3) to (– 9, – 3)
Orientation: vertical
2 − −8
𝑎=
2
𝑎=5
−6 + −6 2 + −8
=
,
2
2
Center ℎ, 𝑘
𝑥−ℎ
𝑏2
2
𝑥 − −6
32
𝑥+6
9
2
𝑏=
𝑦−𝑘
+
𝑎2
2
2
=1
𝑦 − −3
+
52
𝑦+3
+
25
2
=1
2
=1
−3 − −9
2
= −6, −3
𝑏 =3
Example 2b: Write an equation for an ellipse with given
characteristics.
vertices at(– 4, 4) and (6, 4); foci at (– 2, 4) and (4, 4)
Orientation: horizontal
6 − −4
𝑎=
2
𝑐 2 = 𝑎2 − 𝑏2
32
=
52
−
𝑏2
𝑎=5
𝑐=
4 − −2
2
Center ℎ, 𝑘
=
𝑐=3
−4 + 6 4 + 4
,
2
2
𝑏2 = 52 − 32
𝑏2 = 16
𝑏=4
𝑥−ℎ
𝑎2
2
𝑦−𝑘
+
𝑏2
2
𝑥−1
52
𝑥−1
25
2
𝑦−4
+
42
2
𝑦−4
+
16
2
=1
=1
2
=1
= 1, 4
Example 3: Determine the eccentricity of the ellipse given by
𝑥−6 2
100
+
𝑦+1 2
9
= 1.
𝑎 = 100 = 10
𝑐 = 100 − 9 = 91
𝑐
𝑒=
𝑎
91
𝑒=
10
𝑒 ≈ 0.95
The eccentricity is about 0.95, so the
ellipse will appear stretched.
Example 5a: Write the equation in standard form. Identify the
related conic.
𝑥 2 − 6𝑥 − 2𝑦 + 5 = 0
𝑥 2 − 6𝑥 − 2𝑦 = −5
2
𝑥 − 6𝑥
= 2𝑦 − 5
𝑥 2 − 6𝑥 + 9 = 2𝑦 − 5 + 9
𝑥−3
𝑥−3
2
2
= 2𝑦 + 4
=2 𝑦+2
The conic section is a parabola
with vertex (3, – 2).
2
𝑏
−6
=
2
2
2
= −3
2
=9
Example 5b: Write the equation in standard form. Identify the
related conic.
𝑥 2 + 𝑦 2 − 12𝑥 + 10𝑦 + 12 = 0
𝑥 2 − 12𝑥
+ 𝑦 2 + 10𝑦
= −12
𝑥 2 − 12𝑥 + 36 + 𝑦 2 + 10𝑦 + 25 = −12 +36 + 25
𝑥−6
2
+ 𝑦+5
2
= 49
The conic section is a circle with
center (6, – 5) and radius 7.
Example 5c: Write the equation in standard form. Identify the
related conic.
𝑥 2 + 4𝑦 2 − 6𝑥 − 7 = 0
𝑥 2 − 6𝑥
+ 4𝑦 2 = 7
𝑥 2 − 6𝑥 + 9 + 4𝑦 2 = 7 + 9
𝑥−3
2
+ 4𝑦 2 = 16
𝑥−3
16
2
4𝑦 2 16
+
=
16
16
𝑥 − 3 2 𝑦2
+
=1
16
4
The conic section is an ellipse
with center (3, 0).
−1
math!
7-2 Assignment: TX p438, 4-36 EOE + 34