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So far, we ve described various surfaces by adding squared binomials to each other You have every right to ask, What will happen if we subtract any of the squared binomials in equations like these We ll do that shortly, and you ll see a few examples of what can take place When we add squared binomials, the graphs always turn out to be spheres, oblate spheres, ellipsoids, or oblate ellipsoids in Cartesian xyz space These are closed surfaces They re air-tight If we subtract one or more of the squared binomials, we get open surfaces that can t hold air Such surfaces can take diverse, interesting forms
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Consider a distorted sphere represented by the following equation: 12x2 + 72x + 20y2 80y + 15z2 30z = 143 What are the coordinates of the center What are the axial radii Is the object an oblate sphere, an ellipsoid, or an oblate ellipsoid
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This problem requires a lot of insight to solve! Let s begin by adding 203 to each side of the equation to obtain 12x2 + 72x + 20y2 80y + 15z2 30z + 203 = 60 The number we ve added, 203, happens to be the sum of 108, 80, and 15 Let s add these three numbers into the above equation just after the terms 72x, 80y, and 30z, respectively The equation then becomes 12x2 + 72x + 108 + 20y2 80y + 80 + 15z2 30z + 15 = 60 Grouping the addends on the left-hand side by threes gives us (12x2 + 72x + 108) + (20y2 80y + 80) + (15z2 30z + 15) = 60 which is equivalent to 12(x2 + 6x + 9) + 20(y2 4y + 4) + 15(z2 2z + 1) = 60 The three trinomials factor into perfect squares, so we can further morph the equation to obtain 12(x + 3)2 + 20(y 2)2 + 15(z 1)2 = 60 Dividing through by 60, we get (x + 3)2/5 + (y 2)2/3 + (z 1)2/4 = 1
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We recall that general formula for a distorted sphere in Cartesian xyz space is (x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where (x0,y0,z0) are the coordinates of the center, a is the axial radius in the x direction, b is the axial radius in the y direction, and c is the axial radius in the z direction In this situation, we have (x0,y0,z0) = ( 3,2,1) a = 51/2 b = 31/2 c = 41/2 = 2 Our object is an oblate ellipsoid centered at ( 3,2,1) The radius in the x direction is 51/2 The radius in the y direction is 31/2 The radius in the z direction is 2
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Let s look at three general objects that arise in Cartesian xyz space from equations with sums and differences of terms containing x2, y2, and z2
Hyperboloid of one sheet Figure 17-11 shows a generic example of a hyperboloid of one sheet In this context, the term sheet refers to an unbroken surface We get this type of object when we graph an equation of the form
x2/a2 + y2/b2 z2/c2 = 1 where a, b, and c are positive real-number constants This equation is like the one for a distorted sphere, except that one of the plus signs has been replaced by a minus sign That sign change makes a huge difference! Instead of a closed surface centered at the origin, we get an infinitely tall, pinched cylinder whose axis lies along the coordinate z axis, and whose center coincides with the origin The dimensions and shape of the hyperboloid depend on the values of a, b, and c The perpendicular cross sections are always circles or ellipses If we move the minus sign so that it s in front of the term containing y2 instead of the term containing z2, we get the general equation x2/a2 y2/b2 + z2/c2 = 1 Again, we get a hyperboloid of one sheet, but its axis is along the coordinate y axis, and its center is at the origin If we move the minus sign one more place to the left, putting it in front of the term containing x2, the general equation becomes x2/a2 + y2/b2 + z2/c2 = 1 This is the general form of the equation for a hyperboloid of one sheet whose axis coincides with the coordinate x axis, and whose center is at the origin
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