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Ellipsoid-plane intersection Let E be the ellipsoidx2/9 +
Chapter 12, Problem 88AE(choose chapter or problem)
Ellipsoid-plane intersection Let E be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1\), P be the plane z=Ax+By, and C be the intersection of E and P.
(a) Is C an ellipse for all values of A and B? Explain.
(b) Sketch and interpret the situation in which A=0 and \(B \neq 0\).
(c) Find an equation of the projection of C on the xy-plane.
(d) Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2}\). Find a parametric description of C as a curve in \(\mathbf{R}^{3}\). (Hint: Assume C is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find a, b, c, d, e, and f.)
Questions & Answers
QUESTION:
Ellipsoid-plane intersection Let E be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1\), P be the plane z=Ax+By, and C be the intersection of E and P.
(a) Is C an ellipse for all values of A and B? Explain.
(b) Sketch and interpret the situation in which A=0 and \(B \neq 0\).
(c) Find an equation of the projection of C on the xy-plane.
(d) Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2}\). Find a parametric description of C as a curve in \(\mathbf{R}^{3}\). (Hint: Assume C is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find a, b, c, d, e, and f.)
ANSWER:Solution 88AE
a.
Consider the ellipsoid
(1)
And the plane
(2)
To find the intersection curve substitute (2) into (1)
Look a general conic section of the form
is an ellipse if
Therefore the curve C is an ellipse.
b.
Consider therefore the equation of the is
It’s an ellipse having major axis and minor axis is
Sketch the graph considering therefore the equation of
To draw a graph by using maple here follows the steps given below:
Step1: Write the equations in the maple spread sheet as shown below:
