   Chapter 10.1, Problem 41E

Chapter
Section
Textbook Problem

If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. Then eliminate the parameter and identify the curve. To determine

To find: The parametric equation of the cycloid for the variables x and y and eliminate the parameter and identify the curve.

Explanation

Calculation:

Consider the geometry of the circle shown below in Figure 1.

From Figure 1, radius of the inner circle |OS| is b and radius of the outer circle |OR| is a and length of the segment, |RS|=ab.

In triangle SOT,

Length of the bottom side,

|OT|=acosθx=αcosθ (1)

Length of the vertical side, |ST|=asinθ

In triangle SRP,

Length of the bottom side,|SP|=(absinθ)

Length of the diagonal,

|PT|=STSP=asinθ(ab)sinθ=asinθasinθ+bsinθy=bsinθ (2)

The co-ordinates of point P (αcosθ,bsinθ)

From equations (1) and (2), The parametric equation for th

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