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13.6.Ex42-44 - 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) 409 Combining (3) and (4) gives u 1 = 0 , y ( t 0 ), c 2 n b 2 r y ( t 0 ) = y ( t 0 ) 0 , 1 , c 2 n b 2 r u 2 = x ( t 1 ), 0 , c 2 m a 2 r x ( t 1 ) = x ( t 1 ) 1 , 0 , c 2 m a 2 r u 3 = x ( t 2 ), b 2 m a 2 n x ( t 2 ), 0 = x ( t 2 ) 1 , b 2 m a 2 n , 0 (c) Now, to show that the tangent lines are contained in P , we must show that their direction vectors are in P , that is, they are orthogonal to the vector n = m a 2 , n b 2 , r c 2 normal to the plane. We show this by proving that the following dot products are zero. That is, u 1 · n = y ( t 0 ) 0 , 1 , c 2 n b 2 r · m a 2 , n b 2 , r c 2 = y ( t 0 ) n b 2 n b 2 = 0 u 2 · n = x ( t 1 ) 1 , 0 , c 2 m a 2 r · m a 2 , n b 2 , r c 2 = x ( t 1 ) m a 2 m a 2 = 0 u 3 · n = x ( t 2 ) 1 , b 2 m a 2 n , 0 · m a 2 , n b 2 , r c 2 = x ( t 2 ) m a 2 m a 2 = 0 Since the tangent lines intersect P (at the point P ) and their direction vectors are in P , it follows that the tangent lines are contained in the plane P . 42. Let S be the hyperboloid x 2 + y 2 = z 2 + 1 and let P = ( α , β , 0 ) be a point on S in the ( x , y ) -plane. Show that there are precisely two lines through P entirely contained in S (Figure 17). Hint: Consider the line r ( t ) = α + at , β + bt , t through P . Show that r ( t ) is contained in S if ( a , b ) is one of the two points on the unit circle obtained by rotating ( α , β ) through ± π 2 . This proves that a hyperboloid of one sheet is a
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