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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

4. x = 2 cos θ, y = 1 + sin θ

To determine

To Find:

Derive the Cartesian equation of the curve and sketch the graph of the parametric equation.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=2cosθ (1)

The parametric equation for the variable y is as follows.

y=1+sinθ (2)

Calculation:

Eliminate the parameter t as below.

Rearrange the equation (1) as,

cosθ=x2 (3)

Rearrange the equation (2) as,

sinθ=y1 (4)

Use the relation for right angled triangle

sin2θ+cos2θ=1 (5)

Substitute equation (3) and (4) in (5),

(x2)2+(y1)2=1x24+(y1)2=1

(x0)222+(y1)212=1 (6)

The ellipse equation is as below.

(xh)2a2+(yk)2b2=1 (5)

Here, the point (h,k) is the center point, a is the semi major axis, b is the semi minor axis

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