Question 1

Which of the following is the correct parametric equation for a circle with center at $$(-3, 2)$$ and radius $$4$$?

Accepted Answer

$$x = -3 + 4 \cos t, \quad y = 2 + 4 \sin t$$

Question 2

Given the parametric equations $$x = -3 + 4 \cos t$$ and $$y = 2 + 4 \sin t$$, what is the Cartesian equation of the circle?

Accepted Answer

$$(x + 3)^2$ + (y - 2)^2$ = 16$$

Question 3

If a circle is described by the parametric equations $$x = h + r \cos t$$, $$y = k + r \sin t$$, which of the following statements is true?

Accepted Answer

The center of the circle is at $$(h, k)$$ and the radius is $$r.$$

Question 4

Which parametric equations describe an ellipse centered at $$(-3, 2)$$ with vertices at $$(-3, 7)$$ and $$(-3, -3)$$?

Accepted Answer

Question 5

Given the ellipse with center $$(-3, 2)$$, vertices at $$(-3, 7)$$ and $$(-3, -3)$$, and points at $$(1, 2)$$ and $$(7, 2)$$, what are the lengths of the major and minor axes?

Accepted Answer

Major axis: $$10$$, Minor axis: $$6$$

Question 6

Which of the following parametric equations correctly describes the ellipse with center $$(-3, 2)$$, major axis length $$10 ($$vertical), and minor axis length $$6 ($$horizontal)?

Accepted Answer

$$x = -3 + 3 \cos t, \quad y = 2 + 5 \sin t$$

Question 7

A circle is described by $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t. $$What is the range of values for $$t to $$complete one full revolution?

Accepted Answer

$$0 \leq t \u003c 2\pi$$

Question 8

If the parametric equations for a circle are $$x = h + r \cos t$$, $$y = k + r \sin t$$, what is the effect of increasing $$r$$?

Accepted Answer

The circle's radius increases, making the circle larger.

Question 9

Given the parametric equations $$x = -3 + 3 \cos t$$, $$y = 2 + 5 \sin t$$ for an ellipse, what is the Cartesian equation of the ellipse?

Accepted Answer

$$\frac{(x + 3)^2$}{9} + \frac{(y - 2)^2$}{25} = 1$$

Question 10

A medical imaging device traces a circular path with center $$(-3, 2)$$ and radius $$4. If $$the device starts at the point $$(1, 2)$$, what is the value of $$t in $$the parametric equations $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t$$?

Accepted Answer

$$t = 0$$

Question 11

An engineer needs to design an elliptical track with center $$(-3, 2)$$, vertical major axis of length $$10$$, and horizontal minor axis of length $$6. $$Which parametric equations should be used?

Accepted Answer

$$x = -3 + 3 \cos t, \quad y = 2 + 5 \sin t$$

Question 12

Which of the following points lies on the circle defined by $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t$$ for some $$t$$?

Accepted Answer

$$(1, 2)$$

Question 13

A runner completes one lap around a circular track described by $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t. $$How far does the runner travel?

Accepted Answer

$$8\pi$$

Question 14

If the parametric equations for an ellipse are $$x = h + a \cos t$$, $$y = k + b \sin t$$, what do $$a$$ and $$b$$ represent?

Accepted Answer

$$a is $$half the length of the horizontal axis, $$b is $$half the length of the vertical axis.

Question 15

A circle with center $$(-3, 2)$$ and radius $$4 is $$traced by a robot arm. If the arm is at position $$(x, y)$$, which equation must be satisfied?

Accepted Answer

$$(x + 3)^2$ + (y - 2)^2$ = 16$$

Question 16

Which parametric equations describe an ellipse with center $$(-3, 2)$$, vertical major axis of length $$10$$, and horizontal minor axis of length $$6$$?

Accepted Answer

$$x = -3 + 3 \cos t, \quad y = 2 + 5 \sin t$$

Question 17

If $$t$$ increases from $$0 to$$ $$2\pi in $$the parametric equations $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t$$, what geometric path is traced?

Accepted Answer

A full circle centered at $$(-3, 2)$$ with radius $$4.$$

Parametric Equations for Circles and Ellipses

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