Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Graph the ellipse r=5/2+cosθ, and obtain the standard form of its Cartesian representation.
If e<1, the equation r=e d1+e cosθ defines an ellipse.
Writing the given equation as
the values e=1/2 and d=5 can be immediately deduced. Hence, it follows that
use plots in
p1:=display(implicitplot(r = 5/(2+cos(t)),r=0..5,t=-Pi..Pi,coords=polar,gridrefine=3,color=red),labels=[x,y]):
Figure 7.1.12(a) Ellipse r=5/2+cosθ
Figure 7.1.12(a), a graph of this ellipse, includes the center C:−5/3,0 as the blue dot, and (as green dots) the two foci: F1 at the origin, and F2 at −10/3,0. The standard form of the Cartesian representation of this ellipse, namely,
can be deduced from the figure and the computed values of a,b, and c.
Figure 7.1.12(b) contains the relevant portion of the
tutor in which a graph and significant details of a conic are obtained.
The graph in Figure 7.1.12(b) has been modified by selecting Constrained Scaling. The vertical line on the right is the directrix, whose equation is x=5.
The standard form for the Cartesian representation is provided, along with the values of the parameters a,b,h,k, where h,k is the center of the ellipse.
Note that the values of a,b, and e agree with those found in the Mathematical Solution. The latus rectum is the length of a focal chord perpendicular to the major axis.
The tutor, housed in the Student Precalculus package, can be obtained from the Tools/Tutors menu.
Figure 7.1.12(b) Conic Sections tutor
Figure 7.1.12(a), or the graph in Figure 7.1.12(b), can be obtained interactively with the Plot Builder, or with the following command. (Select Evaluate in the Context Panel.)
The direct algebraic conversion of the polar form of the ellipse to the standard form of the Cartesian representation is a tedious affair, made somewhat easier with two of Maple's tools, namely, the Context Panel and the Equation Manipulator (an Assistant).
Enter the polar form of the ellipse.
Press the Enter key.
Context Panel: Evaluate at a Point
r = sqrt(x^2+y^2)
theta = arctan(y,x)
Context Panel: Cross Multiply
See Figure 7.1.12(c)
→evaluate at point
Figure 7.1.12(c) The Equation Manipulator
Table 7.1.12(a) lists the steps to perform in the Equation Manipulator.
Multiply equation by 1/x2+y2
Add −x to both sides
Square both sides
Group terms on left side
Apply simplify to left side
Complete the square on the left side
Add 100/3 to equation
Multiply equation by 3/100
Table 7.1.12(a) Application of the Equation Manipulator
At this point, only pencil-and-paper suffices to obtain the final form, namely, x+5/32100/9+y225/3=1.
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