   Chapter 10.1, Problem 43E

Chapter
Section
Textbook Problem

# A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as x = 2 a cot θ y = 2 a sin 2 θ Sketch the curve. To determine

To find: The parametric equation for the curve given as x=2acotθ and y=2asin2θ .

Explanation

Given data:

The curve MariaAgnesi is shown in below figure1.

Calculation:

The horizontal distance of point C is x .

xOC=cosθ (1)

The vertical distance of point A is y .

yOA=sinθ (2)

The sum of all angles forming triangle OAD will be equal to 180 .

φ+θ=90φ=90θφ+90+α=18090θ+90+α=180α=θ

OA2a=sinθOA=sinθ2a

Substitute (sinθ2a) for OA in equation (2),

yOA=sinθ

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