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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

In the figure the circle of radius a is stationary, and for every θ, the point P is the midpoint of the segment QR. The curve traced out by P for 0 < θ < π is called the long-bow curve. Find parametric equations for this curve.

Chapter 10, Problem 57RE, In the figure the circle of radius a is stationary, and for every , the point P is the midpoint of

To determine

To find: The parametric equation of the given curve.

Explanation

Given:

The polar equation of the circle is as follows.

r=2asinθ (1)

The polar equation for the variable x is as follows.

x=rcosθ (2)

The polar equation for the variable y is as follows.

y=rsinθ (3)

Calculation:

Substitute equation (1) in (2) we get,

x=2asinθcosθ (4)

Substitute equation (1) in (3) we get,

y=2asin2θ (5)

The coordinate of point Q is (x1,y1)=(2asinθcosθ,2asin2θ) .

Determine the coordinates of point R.

From the diagram given in the given in the question,

y=2a (6)

Substitute equation (3) in (6) to get the below equation.

2a=rsinθ

r=2asinθ (7)

Substitute equation (7) in (2).

x=2asinθcosθ

x=2acotθ (8)

The coordinate of point R is (x2,y2)=(2acotθ,2a)

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