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It s reasonable to ask, What if the center of a hyperboloid, or the vertex of an elliptic cone, lies somewhere other than the origin, say at (x0,y0,z0) What happens to the equation in that case If you re willing to exercise your mathematical intuition, you can probably guess the answer Make the following substitutions in the equation: Replace every occurrence of x with x x0 Replace every occurrence of y with y y0 Replace every occurrence of z with z z0 Consider a hyperboloid of two sheets such as the one in Fig 17-12 The straight-line axis of the bowls lies along the coordinate y axis, and the center of the entire object is at the origin If a = 2, b = 3, and c = 4, the equation is x2/22 + y2/32 z2/42 = 1
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which simplifies to x2/4 + y2/9 z2/16 = 1 Now suppose that you change the equation to (x 7)2/4 + (y + 1)2/9 (z 5)2/16 = 1 You ve moved the entire hyperboloid, without altering its overall shape or orientation It has a new center whose coordinates are (7, 1,5) instead of (0,0,0) The straight-line axis of the two bowls is parallel to, but no longer coincides with, the coordinate y axis If you want to disguise this equation, you can multiply it through by the product of the denominators on the left-hand side, getting 144(x 7)2 + 64(y + 1)2 36(z 5)2 = 576
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Consider the object in Cartesian xyz space represented by 3x2 + 6x 4y2 16y + 2z2 4z = 35 What do we get when we graph this equation in Cartesian xyz space Where is the center of the object How is its axis oriented
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As with some of the examples we ve seen, we need lot of intuition to solve this problem Let s subtract 11 from both sides of the equation That gives us 3x2 + 6x 4y2 16y + 2z2 4z 11 = 24 When we subtract 11, we in effect add 11, which happens to be the sum of 3, 16, and 2 That means we can rewrite the above equation as 3x2 + 6x 4y2 16y + 2z2 4z + 3 16 + 2 = 24 which can be rearranged to get 3x2 + 6x + 3 4y2 16y 16 + 2z2 4z + 2 = 24 Grouping the terms on the left-hand side into trinomials, and paying special attention to the signs associated with the variable y as we group the second three terms, we get (3x2 + 6x + 3) (4y2 + 16y + 16) + (2z2 4z + 2) = 24 which morphs to 3(x2 + 2x + 1) 4(y2 + 4y + 4) + 2(z2 2z + 1) = 24
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and further to 3(x + 1)2 4(y + 2)2 + 2(z 1)2 = 24 Dividing through by 24, we get (x + 1)2/8 (y + 2)2/6 + (z 1)2/12 = 1 This is the equation of a hyperboloid of one sheet whose center is at ( 1, 2,1), and whose axis is oriented along a line parallel to the coordinate y axis
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Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Suppose that a plane contains the point (0,0,0), and the standard form of a vector normal to the plane is 4i + 4j 4k Find the plane s equation in standard form 2 Suppose that a plane contains the point (4,5,6), and the standard form of a vector normal to the plane is 2i + 0j + 0k Find the plane s equation in standard form 3 Consider a sphere whose equation is x2 + 2x + 1 + y2 2y + 1 + z2 + 8z + 16 = 64 What are the coordinates of the center of this sphere What s its radius 4 What s the equation of a sphere centered at the point (5,7, 3) and whose radius is equal to the positive square root of 23 5 Consider the equation 8(x 1)2 + 8(y + 2)2 + 6(z + 7)2 = 24 What sort of object does this equation describe Does the object have a center If so, what are the coordinates of the center point Does the object have axial radii If so, what are they 6 Consider the equation 400(x + 2)2 + 225(y 4)2 + 144z2 3\600 = 0 What sort of object does this equation describe Does the object have a center If so, what are the coordinates of the center point Does the object have axial radii If so, what are they
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