# Equations of Ellipses and Hyperbolas

## Presentation on theme: "Equations of Ellipses and Hyperbolas"— Presentation transcript:

Equations of Ellipses and Hyperbolas
Sec. 8.5b

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

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:

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

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:

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

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:

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:

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 

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:

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

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

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:

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

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

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

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