The graph in an xyz-coordinate system of an equation of form ax2 + by2 + cz2 = 1 in

Chapter 7, Problem 34

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The graph in an xyz-coordinate system of an equation of form ax2 + by2 + cz2 = 1 in which a, b, and c are positive is a surface called a central ellipsoid in standard position (see the accompanying figure). This is the three-dimensional generalization of the ellipse ax2 + by2 = 1 in the xy-plane. The intersections of the ellipsoid ax2 + by2 + cz2 = 1 with the coordinate axes determine three line segments called the axes of the ellipsoid. If a central ellipsoid is rotated about the origin so two or more of its axes do not coincide with any of the coordinate axes, then the resulting equation will have one or more cross product terms. (a) Show that the equation 4 3 x2 + 4 3 y2 + 4 3 z2 + 4 3 xy + 4 3 xz + 4 3 yz = 1 represents an ellipsoid, and find the lengths of its axes. [Suggestion: Write the equation in the form xT Ax = 1 and make an orthogonal change of variable to eliminate the cross product terms.] (b) What property must a symmetric 3 3 matrix have in order for the equation xT Ax = 1 to represent an ellipsoid?