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13.6.Ex33-36

13.6.Ex33-36 - S E C T I O N 13.6 A Survey of Quadric...

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S E C T I O N 13.6 A Survey of Quadric Surfaces (ET Section 12.6) 405 33. Find the equation of the ellipsoid passing through the points marked in Figure 14(A). z y x (A) 6 4 4 2 2 6 z y x (B) 4 4 2 2 FIGURE 14 SOLUTION The equation of an ellipsoid is x a 2 + y b 2 + z c 2 = 1 (1) The x , y and z intercepts are ( ± a , 0 , 0 ) , ( 0 , ± b , 0 ) and ( 0 , 0 , ± c ) respectively. The x , y and z intercepts of the desired ellipsoid are ( ± 2 , 0 , 0 ) , ( 0 , ± 4 , 0 ) and ( 0 , 0 , ± 6 ) respectively, hence a = 2, b = 4 and c = 6. Substituting into (1) we get x 2 2 + y 4 2 + z 6 2 = 1 . 34. Find the equation of the elliptic cylinder passing through the points marked in Figure 14(B). SOLUTION The equation of the elliptic cylinder in the xyz -coordinate system is x a 2 + y b 2 = 1 (1) The x and y intercepts are ( ± a , 0 ) and ( 0 , ± b ) respectively. The x and y intercepts of the desired cylinder are ( ± 2 , 0 ) and ( 0 , ± 4 ) respectively, hence a = 2 and b = 4. Substituting into (1) we obtain the following equation: x 2 2 + y 4 2 = 1 . 35. Find the equation of the hyperboloid shown in Figure 15(A).

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13.6.Ex33-36 - S E C T I O N 13.6 A Survey of Quadric...

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