helppp For which values of h is the intersection of the horizontal plane z = h a

Question

helppp

For which values of h is the intersection of the horizontal plane z = h and the hyperboloid -3x2 - 2y2 + 11 z2 = 1 an ellipse? (Enter your answer using interval notation. Enter EMPTY or 0 for the empty set.) Find the equation of the ellipsoid passing through the points (plusminus9, 0, 0), (0, plusminus5, 0) and (0, 0, plusminus2). (Use the standard coordinate variables x, y, and z.) Find the equation of the vertical elliptic cylinder passing through the points (plusminus9, 0, 0) and (0, plusminus3, 0). (Use the standard coordinate variables x, y, and z.) Find the equation of the hyperboloid shown in the figure below. Find the equation of the quadric surface shown in the figure below where a = 4, b = 9, and c = 7.

Explanation / Answer

-3x^2 - 2y^2 + 11z^2 = 1

since it is given that intersection occurs at z = h hence at z = h we ve

-3x^2 - 2y^2 + 11h^2 = 1

-3x^2 - 2y^2 = -11h^2 + 1

or

3x^2 + 2y^2 = 11h^2 - 1

for ellipse the condition which is applicable is

11h^2 - 1 <= 0

hence

11 h^2 <= 1

or

h^2 <= 1/sqrt 11

hence h value ll lie in between

-1/sqrt 11 <= h <= 1/sqrt 11

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