   Chapter 10.1, Problem 12E

Chapter
Section
Textbook Problem

# (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 12. x = 1 2 cos θ , y = 2 sin θ, 0 ≤ θ ≤ π

(a)

To determine

To find: obtain a Cartesian equation of the curve by eliminating parameter t .

Explanation

Given data:

The parametric equation for the variable x is as follows.

x=12cosθ

2x=cosθ (1)

The parametric equation for the variable y is as follows.

y=2sinθ

12y=sinθ (2)

The range of t is 0 to π .

Calculation:

Square and add both the Equations (1) and (2)

(2x2)2+(12y)2=cos2θ+sin2θ

Put 1 for cos2θ+sin2θ in the above equation.

(4x2)+(14y2)=1x2(12)2+y222=1

The Cartesian equation of the curve is x2(12)2+y222=1

(b)

To determine

To plot: The curve using the parametric equations. x=12cosθ and y=2sinθ .

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