Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. x = -7 cos 2t, y = -7 sin 2t, 0 <= t <= pi

Zoe Oneal 2021-02-08 Answered
Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x=7cos2t,y=7sin2t,0tπ
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Expert Answer

curwyrm
Answered 2021-02-09 Author has 87 answers

Step 1
Given:
The following parametric equations, x=7cos(2t),y=7sin(2t),0tπ.
Step 2
a) To eliminate the parameter to obtain an equation in x and y.
We have,
The following parametric equations,
x=7cos(2t)....(1)
y=7sin(2t)....(2)
Adding equations (1) and (2), we get
x2+y2=(7cos(2t))2+(7sin(2t))2
=49cos2(2t)+49sin2(2t)
=49(cos2(2t)+sin2(2t))
=49(1).By using trigonometricident ify}
=49
An equation in x and y is x2+y2=49.
Step 3
b) To describe the curve and indicate the positive orientation.
We have,
x2+y2=(7)2.
The curve represents a circle at the origin, (0,0) and radius, r=7.
To indicate the positive orientation, we will use the following parametric equations,
x=7cos(2t),y=7sin(2t),0tπ.
The interval given is 0tπ.
Therefore, for t=0,
x=7cos(2(0))=7(cos(0))=7(1)=7
y=7sin(2(0))=7sin(0))=7(0)=0.
Implies the initial point is (−7,0).
Fort=π,
x=7cos(2π)=7(1)=7
y=7sin(2π)=7(0)=0
Implies the end point is (−7,0).
The initial point and the end point are equal, that is, (−7,0).
Hence, the orientation is positive in anticlockwise direction.

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