Understanding Parametric Equations of an Ellipse

The video tutorial explains how to find parametric equations for an ellipse, starting with the Cartesian equation. It covers the concepts of major and minor axes, derives the parametric equations using trigonometric identities, and verifies them. The tutorial concludes by discussing the orientation of the ellipse as traced by the parametric equations.

What is the form of the parametric equations for an ellipse?

• x(t) = a sec(t), y(t) = b csc(t)

• x(t) = a tan(t), y(t) = b cot(t)

• x(t) = a cos(t), y(t) = b sin(t)

• x(t) = a sin(t), y(t) = b cos(t)

In the context of an ellipse, what is the relationship between the major and minor axes?

• The major axis is always shorter than the minor axis.

• The major axis is always perpendicular to the minor axis.

• The major axis is always equal to the minor axis.

• The major axis is always longer than the minor axis.

If the length of the major axis is 10, what is the value of a?

• 20

• 2

• 5

• 10

What trigonometric identity is used to derive the parametric equations of an ellipse?

• sin^2(θ) - cos^2(θ) = 1

• tan^2(θ) + 1 = sec^2(θ)

• sin^2(θ) + cos^2(θ) = 1

• 1 + cot^2(θ) = csc^2(θ)

What happens when you substitute x(t) = 5 cos(t) into the Cartesian equation of the ellipse?

• The equation remains unchanged.

• The equation simplifies to sin^2(t).

• The equation simplifies to cos^2(t).

• The equation becomes undefined.

How do you verify the correctness of the parametric equations?

• By checking if the sum of squares equals zero.

• By substituting and simplifying to show the identity holds.

• By comparing with a circle's equation.

• By using the Pythagorean theorem.

What is the orientation of the ellipse when using the given parametric equations?

• Counterclockwise

• Horizontal

• Vertical

• Clockwise

What are the coordinates of the point on the ellipse when t = 0?

• (0, 5)

• (5, 0)

• (3, 0)

• (0, 3)

What are the coordinates of the point on the ellipse when t = π/2?

• (5, 0)

• (3, 5)

• (0, 3)

• (0, 5)

As t increases, how is the ellipse traced using the parametric equations?

• In a counterclockwise direction

• In a clockwise direction

• In a horizontal direction

• In a vertical direction