Equations of Equations of Ellipses and Ellipses and Hyperbolas Hyperbolas Sec. 8.5b Sec. 8.5b
Jan 03, 2016
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