warm-up 1/09
DESCRIPTION
Warm-Up 1/09. 1. 2. B. 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. - PowerPoint PPT PresentationTRANSCRIPT
Warm-Up 1/091.
2.
B
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
2a
2b
2c
Example 1: Graph the ellipse given by the equation.
h=3
(3 ,β1 )
π=β1π=β36=6
Center :foci: (3 Β±3 β3 ,β1 )
(π₯β3 )2
36+
(π¦+1 )2
9=1
π=β9=3π=β36β9=β27=3 β3Orientation: horizontal
vertices: and
co-vertices: and
major axis : π¦=β1minor axis : π₯=3
β’ FFβ’ β’ β’ β’
β’
β’
Example 2a: Write an equation for an ellipse with given characteristics.major axis from (β 6, 2) to (β 6, β 8); minor axis from (β 3, β 3) to (β 9, β 3)
π=2β (β8 )2
π=β3β (β9 )
2π=5 3
Center ΒΏ (β6+ (β6 )2 ,
2+ (β8 )2 )ΒΏ (β6 ,β3 )
Orientation: vertical
(π₯βh )2
π2+
(π¦βπ )2
π2=1
(π₯ββ6 )2
32+
( π¦ββ3 )2
52=1
(π₯+6 )2
9+
(π¦ +3 )2
25=1
Example 2b: Write an equation for an ellipse with given characteristics.vertices at(β 4, 4) and (6, 4); foci at (β 2, 4) and (4, 4)
π=6β (β4 )2 π=5 π=
4β (β2 )2 π=3
π2=π2βπ232=52βπ2π2=52β32π2=16π=4
Center ΒΏ (β4+62 , 4+42 )ΒΏ (1 ,4 )
Orientation: horizontal
(π₯βh )2
π2+
(π¦βπ )2
π2=1
(π₯β1 )2
52+
( π¦β4 )2
42=1
(π₯β1 )2
25+
( π¦β4 )2
16=1
Example 3: Determine the eccentricity of the ellipse given by.
π=β100=10π=β100β9=β91
π=ππ
π=β9110
πβ0.95
The eccentricity is about 0.95, so the ellipse will appear stretched.
Example 5a: Write the equation in standard form. Identify the related conic.
π₯2β6 π₯β2 π¦+5=0(π₯2β6 π₯ )β2 π¦=β5
(π₯2β6 π₯ )=2 π¦β5 (π2 )2
ΒΏ (β62 )2
ΒΏ (β3 )2ΒΏ9
(π₯2β6 π₯+9 )=2 π¦β5+9(π₯β3 )2=2 π¦+4(π₯β3 )2=2 ( π¦+2 )
The conic section is a parabola with vertex (3, β 2).
Example 5b: Write the equation in standard form. Identify the related conic.
π₯2+ π¦2β12π₯+10 π¦ +12=0
(π₯2β12π₯ )+( π¦2+10 π¦ )=β12
(π₯β6 )2+( π¦+5 )2=49
The conic section is a circle with center (6, β 5) and radius 7.
Example 5c: Write the equation in standard form. Identify the related conic.
π₯2+4 π¦2β6 π₯β7=0(π₯2β6 π₯ )+4 π¦2=7(π₯2β6 π₯+9 )+4 π¦2=7+9
(π₯β3 )2+4 π¦ 2=16(π₯β3 )2
16+ 4 π¦
2
16=16 16
(π₯β3 )2
16+ π¦
2
4=1
The conic section is an ellipse with center (3, 0).
ββ1math!
7-2 Assignment: TX p438, 4-36 EOE + 34