Equations of Equations of Ellipses and Ellipses and HyperbolasHyperbolas

Sec. 8.5bSec. 8.5b

Guided PracticeFind a polar equation for the ellipse with a focus at the poleand the given polar coordinates as the endpoints of itsmajor axis. 1.5,0 and 1, Start with a

diagram!

The general equation:1 cos

ker

e

Substitute in points:

1.51

ke

e

1.5 1.5e ke

and 11

ke

e

1 e ke

Guided PracticeFind a polar equation for the ellipse with a focus at the poleand the given polar coordinates as the endpoints of itsmajor axis. 1.5,0 and 1,

Solve the system:

1.5 1.5e ke 1.5 1.5 1e e

1 e ke

0.2e 1.2ke

The equation: 6 5

1 1 5 cosr

6

5 cos

Guided PracticeFind a polar equation for the ellipse with a focus at the poleand the given polar coordinates as the endpoints of itsmajor axis. 3, 2 and 0.75, 2 Start with a

diagram!

The general equation:1 sin

ker

e

Substitute in points:

31

ke

e

3 3e ke

and 0.751

ke

e

0.75 0.75e ke

Guided PracticeFind a polar equation for the ellipse with a focus at the poleand the given polar coordinates as the endpoints of itsmajor axis.

Solve the system: 3 3 0.75 0.75e e 0.6e 1.2ke

The equation: 6 5

1 3 5 sinr

6

5 3sin

3, 2 and 0.75, 23 3e ke 0.75 0.75e ke

Guided PracticeFind a polar equation for the hyperbola with a focus at thepole and the given polar coordinates as the endpoints of itstransverse axis.

and 6, 2 2,3 2 Start with adiagram!

The general equation:1 sin

ker

e

Substitute in points:

61

ke

e

6 6e ke

and 21

ke

e

2 2e ke

Guided PracticeFind a polar equation for the hyperbola with a focus at thepole and the given polar coordinates as the endpoints of itstransverse axis.

and 6, 2 2,3 26 6e ke 2 2e ke

Solve the system: 6 6 2 2e e 2e 6ke

The equation:6

1 2sinr

Analyzing a ConicAnalyzing a Conic

Analyze the conic section given by the equation below. Include in the analysis the values of e, a, b, and c.

16

5 3cosr

Divide numerator anddenominator by 5:

3.2

1 0.6cosr

Eccentricity e = 0.6 It’s an ellipse!!!

Next, graph by hand, andidentify the vertices… 8,0 , 2,Vertices:

Analyzing a ConicAnalyzing a Conic

Analyze the conic section given by the equation below. Include in the analysis the values of e, a, b, and c.

16

5 3cosr

8,0 , 2,Vertices:

So, what is the value of a? 5a How do we find c?

Use the graph 3c Use the definition of eccentricity 3c

Analyzing a ConicAnalyzing a Conic

Analyze the conic section given by the equation below. Include in the analysis the values of e, a, b, and c.

16

5 3cosr

Finally, what is the value of b?

2 2b a c

Pythagorean relation for an ellipse:2 2 2a b c

25 9 4

Analyzing a ConicAnalyzing a Conic

Analyze the conic section given by the equation below. Include in the analysis the values of e, a, b, and c.

16

5 3cosr

Using all of this information, we canwrite the Cartesian equation of this ellipse

2 2

2 21

x h y k

a b

2 231

25 16

x y

Whiteboard PracticeFind a polar equation for the hyperbola with a focus at thepole and the given polar coordinates as the endpoints of itstransverse axis.

and 3,0 1.5, Start with adiagram!

The general equation:1 cos

ker

e

Substitute in points:

31

ke

e

3 3e ke

and 1.51

ke

e

1.5 1.5e ke

Whiteboard PracticeFind a polar equation for the hyperbola with a focus at thepole and the given polar coordinates as the endpoints of itstransverse axis.

and 3,0 1.5,

Solve the system:

3 3e ke 1.5 1.5e ke 3 3 1.5 1.5e e

3e 6ke

The equation:6

1 3cosr

Whiteboard PracticeWhiteboard Practice

Graph the given conic, and find the values of e, a, b, and c.

ee = 5/6, = 5/6, aa = 6, = 6, bb = 11, = 11, cc = 5 = 5

11

6 5sinr

11 6

1 5 6 sin

The graph?

Whiteboard PracticeWhiteboard Practice

Graph the given conic, and find the values of e, a, b, and c.

ee = 3/5, = 3/5, aa = 5, = 5, bb = 4, = 4, cc = 3 = 3

16

5 3cosr

16 5

1 3 5 cos

The graph?

Whiteboard PracticeWhiteboard Practice

Graph the given conic, and find the values of e, a, b, and c.

ee = 5, = 5, aa = 1/2, = 1/2, bb = 6 , = 6 , cc = 5/2 = 5/2

12

1 5sinr

The graph?

Whiteboard PracticeWhiteboard Practice

Determine a Cartesian equation for the given polar equation.

6

1 2cosr

Hyperbola:

2 241

4 12

x y