1. Equations of Ellipses and Hyperbolas
Sec. 8.5b

2. Guided Practice
Find a polar equation for the ellipse with a focus at the pole
and the given polar coordinates as the endpoints of its
major axis.
Start with a
diagram!
and
The general equation:
Substitute in points:
and

3. Guided Practice
Find a polar equation for the ellipse with a focus at the pole
and the given polar coordinates as the endpoints of its
major axis.
and
Solve the system:
The equation:

4. Guided Practice
Find a polar equation for the ellipse with a focus at the pole
and the given polar coordinates as the endpoints of its
major axis.
Start with a
diagram!
and
The general equation:
Substitute in points:
and

5. Guided Practice
Find a polar equation for the ellipse with a focus at the pole
and the given polar coordinates as the endpoints of its
major axis.
and
Solve the system:
The equation:

6. Guided Practice
Find a polar equation for the hyperbola with a focus at the
pole and the given polar coordinates as the endpoints of its
transverse axis.
Start with a
diagram!
and
The general equation:
Substitute in points:
and

7. Guided Practice
Find a polar equation for the hyperbola with a focus at the
pole and the given polar coordinates as the endpoints of its
transverse axis.
and
Solve the system:
The equation:

8. Analyzing a Conic
Analyze the conic section given by the equation below.
Include in the analysis the values of e, a, b, and c.
Divide numerator and
denominator by 5:
Eccentricity e = 0.6
 It’s an ellipse!!!
Next, graph by hand, and
identify the vertices…
Vertices:

9. Analyzing a Conic
Analyze the conic section given by the equation below.
Include in the analysis the values of e, a, b, and c.
Vertices:
So, what is the value of a?
How do we find c?
Use the graph 
Use the definition of eccentricity 

10. Analyzing a Conic
Analyze the conic section given by the equation below.
Include in the analysis the values of e, a, b, and c.
Finally, what is the value of b?
Pythagorean relation for an ellipse:

11. Analyzing a Conic
Analyze the conic section given by the equation below.
Include in the analysis the values of e, a, b, and c.
Using all of this information, we can
write the Cartesian equation of this ellipse

12. Whiteboard Practice
Find a polar equation for the hyperbola with a focus at the
pole and the given polar coordinates as the endpoints of its
transverse axis.
Start with a
diagram!
and
The general equation:
Substitute in points:
and

13. Whiteboard Practice
Find a polar equation for the hyperbola with a focus at the
pole and the given polar coordinates as the endpoints of its
transverse axis.
and
Solve the system:
The equation:

14. Whiteboard Practice e = 5/6, a = 6, b = 11, c = 5
Graph the given conic, and find the values of e, a, b, and c.
The graph?
e = 5/6, a = 6, b = 11, c = 5

15. Whiteboard Practice e = 3/5, a = 5, b = 4, c = 3
Graph the given conic, and find the values of e, a, b, and c.
The graph?
e = 3/5, a = 5, b = 4, c = 3

16. Whiteboard Practice e = 5, a = 1/2, b = 6 , c = 5/2
Graph the given conic, and find the values of e, a, b, and c.
The graph?
e = 5, a = 1/2, b = 6 , c = 5/2

17. Whiteboard Practice
Determine a Cartesian equation for the given polar equation.
Hyperbola: