Parametric Equations Worksheet

**Page 1**

1.

John hits a ball when it is 3 ft above the ground . The ball leaves the bat with an initial velocity 20 ft/sec, at 60° angle with the horizontal and heads towards a 9 ft wall at a distance of 8 ft . At what height does the ball hit the wall?

a. | 9 ft | ||

b. | 8 ft | ||

c. | 6.6 ft | ||

d. | 8.6 ft |

The motion of the ball can be given by the parametric equations

Time taken by the ball to reach the 9 ft wall, which is at 8 ft from the home place =

At 0.8 sec,

[Substitute

So, the ball will hit the 9 ft wall at a height of 6.6 ft .

Correct answer : (3)

2.

Find the equation obtained by eliminating the parameter $t$ from the parametric equations $x$ = $t$ + 1, $y$ = 2$t$ + 5 .

a. | $y$ = 2$x$ + 3 | ||

b. | $y$ = 3$x$ + 2 | ||

c. | $y$ = 3$x$ - 2 | ||

d. | $y$ = 2$x$ - 3 |

[Solve for

Then

[Substitute

So,

Correct answer : (1)

3.

The parametric equations of the path of a moving particle are $x$ = 8$t$ + 8, $y$ = 9$t$ - 2. Find the path of the particle.

a. | an ellipse | ||

b. | a straight line | ||

c. | a circle | ||

d. | a parabola |

[Write the parametric equations of the path of the particle.]

Consider

[Solve for

[Substitute

So,

The equation

So, the path of the moving particle is a straight line.

Correct answer : (2)

4.

Find the equation obtained by eliminating the parameter $t$ from the equations $x$ = $\sqrt{25-{t}^{2}}$, $y$ = $t$ .

a. | $x$ ^{2} - $y$^{2} = 25 | ||

b. | $x$ ^{2} + $y$^{2} = 25 | ||

c. | $x$ ^{2} - $y$ = 25 | ||

d. | $x$ ^{2} + $y$ = 25 |

[Substitute

[Square on both sides.]

[Simplify.]

So,

Correct answer : (4)

5.

Choose the equation obtained by eliminating the parameter $t$ from the equations $x$ = 8 cos $t$, $y$ = 8 sin $t$, 0 ≤ $t$ ≤ $\pi $.

a. | $x$ ^{2} + $y$^{2} = 64 | ||

b. | $x$ ^{2} + $y$^{2} = 8 | ||

c. | $x$$y$ = 64 | ||

d. | $x$ ^{2} - $y$^{2} = 64 |

cos

[Solve for cos

[Use the identity cos

[Simplify.]

So,

Correct answer : (1)

6.

Choose the equation obtained by eliminating the parameter $t$ from the equations $x$ = $t$ + 5, $y$ = $\frac{16}{t}$.

a. | ($x$ + 5)$y$ = 16 | ||

b. | ($x$ - 5)$y$ = 16 | ||

c. | ($x$ - 16)$y$ = 8 | ||

d. | ($x$ + 16)$y$ = 5 |

[Solve for

[Substitute

(

[Simplify.]

So, the required equation obtained by eliminating the parameter

Correct answer : (2)

7.

Identify the equation obtained by eliminating the parameter $t$ from the equations $x$ = 2$t$, $y$ = 8 - $t$^{2}.

a. | $x$ ^{2} - 4$y$ + 32 = 0 | ||

b. | $x$ ^{2} + 4$y$ = 0 | ||

c. | $x$ ^{2} + 4$y$ - 32 = 0 | ||

d. | $x$ ^{2} - 4$y$ = 0 |

[Solve for

[Substitute

[Simplify.]

So,

Correct answer : (3)

8.

Find the equation obtained by eliminating the parameter from the equations $x$ = $a$$t$^{2}, $y$ = 2$a$$t$.

a. | $x$ ^{2} + 4$a$$y$ = 0 | ||

b. | $y$ ^{2} = 4$a$$x$ | ||

c. | $y$ ^{2} + 4$a$$x$ = 0 | ||

d. | $x$ ^{2} = 4$a$$y$ |

Consider

[Square on both sides.]

[Substitute

So,

Correct answer : (2)

9.

What is the graph represented by $x$ = 3 + 5$t$, $y$ = 4 + 5$t$, 1 ≤ $t$ ≤ 3?

a. | a straight line | ||

b. | a line | ||

c. | a circle | ||

d. | a parabola |

The end values of

[As 1 ≤

At

[Substitute

At

[Substitute

So, the end points of the graph are (8, 9) and (18, 19).

[Solve for

[Substitute

[Simplify.]

So,

As the graph has end points at (8, 9), (18, 19), it is a straight line.

Correct answer : (1)

10.

Which of the following points corresponds to $t$ = 1 in the parametrization $x$ = $t$^{2} + 4, $y$ = $t$ + ($\frac{2}{t}$), where $t$ is a non zero real number?

a. | (3, 5) | ||

b. | (5, - 3) | ||

c. | (- 3, 5) | ||

d. | (5, 3) |

At

[Substitute

At

[Substitute

So, (5, 3) is a point corresponding to

Correct answer : (4)