Ellipse Properties and Polar Equations

This video tutorial explores polar equations of conic sections, focusing on graphing an ellipse. It begins with a review of polar equations and conic sections, then provides a detailed example of graphing an ellipse. The tutorial covers finding the major axis endpoints, determining the center and minor axis, and completing the ellipse graph. Key concepts such as eccentricity, directrix, and the relationship between a, b, and c in an ellipse are discussed.

What is the directrix of a polar equation with cosine theta?

• Horizontal at y = ±D

• Vertical at x = ±D

• Diagonal at y = x

• No directrix

In the given example, what is the eccentricity of the ellipse?

• 2

• 4

• 12

• 8

What is the value of D in the example given in the video?

• 2

• 4

• 6

• 8

What is the major axis of the ellipse aligned with when the equation contains sin theta?

• X-axis

• No specific axis

• Y-axis

• Z-axis

What is the radius when theta is 3π/2 in the example?

• 2

• 8

• 4

• 6

How do you find the center of the ellipse?

• By averaging the endpoints of the major axis

• By finding the midpoint of the minor axis

• By using the directrix

• By calculating the eccentricity

How far is the center of the ellipse from the endpoints of the major axis?

• 4/3 units

• 8/3 units

• 6 units

• 2 units

What is the relationship between a, b, and c in an ellipse?

• a^2 = b^2 + c^2

• a^2 = b^2 - c^2

• a^2 = b^2 * c^2

• a^2 = c^2 - b^2

What is the approximate length of the minor axis?

• 1.5 units

• 3.5 units

• 2.3 units

• 4.3 units

Where is the second focus of the ellipse located?

• At the origin

• 4/3 units from the vertex

• 8/3 units from the center

• 2 units from the directrix