Circle Equations and Parametric Forms

The video tutorial explains how to convert parametric equations to rectangular form by eliminating the parameter t. It provides general guidelines and examples, including linear, square root, exponential, and trigonometric functions. The tutorial also discusses domain restrictions and graphing the resulting equations.

What is the first step in converting parametric equations to rectangular form?

• Use a trigonometric identity

• Solve one of the equations for y

• Solve one of the equations for t

• Graph the parametric equations

In the first example, what is the rectangular form of the line given by the parametric equations x = 4t + 4 and y = t + 2?

• y = 4x + 1

• y = 1/4x + 1

• y = 2x + 4

• y = x - 2

Why must the domain be restricted in the second example involving x = √t?

• Because t can be any real number

• Because t must be less than zero

• Because t must be greater than or equal to zero

• Because t must be a positive integer

What is the rectangular form of the equation y = t - 5 when x = √t?

• y = x - 5

• y = x^2 - 5

• y = x + 5

• y = x^2 + 5

In the third example, how is the exponential equation x = e^(-t) rewritten to solve for t?

• As a trigonometric equation

• As a natural log equation

• As a square root equation

• As a polynomial equation

What is the rectangular form of the equation y = e^t - 1 when x = e^(-t)?

• y = 1/x - 1

• y = x - 1

• y = x + 1

• y = 1/x + 1

In the final example, which trigonometric identity is used to eliminate the parameter t?

• 1 + cot^2(θ) = csc^2(θ)

• tan^2(θ) + 1 = sec^2(θ)

• sin^2(θ) + cos^2(θ) = 1

• sin(2θ) = 2sin(θ)cos(θ)

What is the rectangular form of the parametric equations x = 3cos(t) and y = 3sin(t)?

• x^2 + y^2 = 9

• x^2 - y^2 = 9

• x^2 + y^2 = 1

• x^2 - y^2 = 1

What is the center of the circle represented by the rectangular equation x^2 + y^2 = 9?

• (9, 9)

• (3, 3)

• (0, 0)

• (1, 1)

What is the radius of the circle represented by the equation x^2 + y^2 = 9?

• 3

• 2

• 4

• 1