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 ,−1 )

𝑘=−1𝑎=√36=6

Center :foci: (3 ±3 √3 ,−1 )

(𝑥−3 )2

36+

(𝑦+1 )2

9=1

𝑏=√9=3𝑐=√36−9=√27=3 √3Orientation: horizontal

vertices: and

co-vertices: and

major axis : 𝑦=−1minor axis : 𝑥=3

• FF• • • •

•

•

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)

𝑎=2− (−8 )2

𝑏=−3− (−9 )

2𝑎=5 3

Center ¿ (−6+ (−6 )2 ,

2+ (−8 )2 )¿ (−6 ,−3 )

Orientation: vertical

(𝑥−h )2

𝑏2+

(𝑦−𝑘 )2

𝑎2=1

(𝑥−−6 )2

32+

( 𝑦−−3 )2

52=1

(𝑥+6 )2

9+

(𝑦 +3 )2

25=1

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)

𝑎=6− (−4 )2 𝑎=5 𝑐=

4− (−2 )2 𝑐=3

𝑐2=𝑎2−𝑏232=52−𝑏2𝑏2=52−32𝑏2=16𝑏=4

Center ¿ (−4+62 , 4+42 )¿ (1 ,4 )

Orientation: horizontal

(𝑥−h )2

𝑎2+

(𝑦−𝑘 )2

𝑏2=1

(𝑥−1 )2

52+

( 𝑦−4 )2

42=1

(𝑥−1 )2

25+

( 𝑦−4 )2

16=1

Example 3: Determine the eccentricity of the ellipse given by.

𝑎=√100=10𝑐=√100−9=√91

𝑒=𝑐𝑎

𝑒=√9110

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.

𝑥2−6 𝑥−2 𝑦+5=0(𝑥2−6 𝑥 )−2 𝑦=−5

(𝑥2−6 𝑥 )=2 𝑦−5 (𝑏2 )2

¿ (−62 )2

¿ (−3 )2¿9

(𝑥2−6 𝑥+9 )=2 𝑦−5+9(𝑥−3 )2=2 𝑦+4(𝑥−3 )2=2 ( 𝑦+2   )

The conic section is a parabola with vertex (3, – 2).

Example 5b: Write the equation in standard form. Identify the related conic.

𝑥2+ 𝑦2−12𝑥+10 𝑦 +12=0

(𝑥2−12𝑥 )+( 𝑦2+10 𝑦 )=−12

(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.

𝑥2+4 𝑦2−6 𝑥−7=0(𝑥2−6 𝑥 )+4 𝑦2=7(𝑥2−6 𝑥+9 )+4 𝑦2=7+9

(𝑥−3 )2+4 𝑦 2=16(𝑥−3 )2

16+ 4 𝑦

2

16=16  16

(𝑥−3 )2

16+ 𝑦

2

4=1

The conic section is an ellipse with center (3, 0).

√−1math!

7-2 Assignment: TX p438, 4-36 EOE + 34