
Concept explainers
a.
Find parametric equations for the set of all points P .
a.

Answer to Problem 34RE
(a−acos2θ,2atanθ−asin2θ)
Explanation of Solution
Given information:
Find parametric equations for the set of all points P determined as shown in the figure such that |OP|=|AB| .(This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissode as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube.)
Calculation:
Consider the figure
Whereas OG&GA are radii of the circle radius a . So OG&GA are congruent and so their opposite angles OAG&AOG are congruent.
Hence OAG=AOG=θ .
For the triangles ABE&ODP , the sides |OP|=|AB| and lines BE‖PD (these lines are perpendicular to the x−axis ) so the angles BAE&POD are congruent.
Hence BAE=POD=θ .
Consider the triangle OBC to find the coordinates of the point B .
Hence OC=2a (diameter of the circle) and BOC=θ , we have
tanθ=BC2aBC=2atanθ
The x -coordinate of the point B is 2a and the y− coordinates is 2atanθ .
Hence the point B=(2a,2atanθ) .
To find the point A , we need to determine the distances AF&OF .
In the triangle OAG , sum of the angles are
θ+θ+AGO=180°AGO=180°−2θ
We know that
AGO+AGF=180°180°−2θ+AGF=180°AGF=2θ
Now consider the triangle AGF, get
sin2θ=AFAG=AFaAF=asin2θ
And
cos2θ=GFAG=GFaGF=acos2θ
The x− coordinate of the point A is
OG+GF=a+acos2θ
The y− coordinate of the point A is AF=asin2θ .
Hence, the point A=(a+cos2θ,asin2θ) .
Find the distance AE&BE:
To get the distance AE , we need to subtract the x -coordinate of A from the x -coordinate of B .
AE=2a−(a+acos2θ)=a−acos2θ
To get the distance BE , we need to subtract the y -coordinate of A from the y -coordinate of B .
BE=2atanθ−asin2θ .
The triangles ABE&OPD are congruent whereas the sides OP,AB are equal and the angles BAE&POD are equal.
Hence, in these two triangles ABE&OPD the corresponding sides AE,OD are equal and BE,PD are equal.
AE=OD=a−acos2θ,BE=PD=2atanθ−asin2θ
Hence, the required coordinates of the point P is (a−acos2θ,2atanθ−asin2θ) .
b.
Use the geometric description of the curve to draw a rough of the curve by hand.
b.

Answer to Problem 34RE
Explanation of Solution
Given information:
Use the geometric description of the curve to draw a rough of the curve by hand. Check your work by using the parametric equations to graph the curve.
Calculation:
Draw the curve represented by the parametric equations
x(θ)=a−acos2θ,y(θ)=2atanθ−asin2θ .
This curve is known as cissoids of Diocles.
The graph of the curve is shown below:
Hence, the result is plotted.
Chapter 1 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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