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