# x1a_sols_2 - Fall 2010 Math 251 Exam 1A Solutions c 2010...

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Fall 2010 Math 251 Exam 1A: Solutions Tue, 28/Sep c ± 2010 Art Belmonte 1. Find the center C and radius r of the sphere 9 x 2 + 9 y 2 + 9 z 2 - 18 x + 6 y - 72 z + 73 = 0. We’ll complete the squares then read off the center and radius. First divide the equation by 9 to ensure the coefﬁcients of the squared terms are 1. x 2 - 2 x + y 2 + 2 3 y + z 2 - 8 z + 73 9 = 0 Complete each square. x 2 - 2 x + 1 + y 2 + 2 3 y + 1 9 + z 2 - 8 z + 16 + 73 9 = 0 + 1 + 1 9 + 16 Write the equation in the standard form for a sphere. ( x - 1 ) 2 + ( y - ( - 1 3 )) 2 +( z - 4 ) 2 = 17 1 9 - 8 1 9 = 9 = 3 2 Now we see that the center is C ( 1 , - 1 3 , 4 ) and that the radius is r = 3. 2. Show that the quadrilateral (four-sided ﬁgure) with corners A ( 2 , - 1 , 1 ) , B ( 5 , 1 , 4 ) , C ( 0 , 1 , 1 ) , D ( 3 , 3 , 4 ) is a parallelogram. Then ﬁnd its area. Let’s form vectors in pairs from opposite sides of the quadrilateral. u = -→ AB = B - A = [ 3 , 2 , 3 ] = D - C = -→ CD v = -→ AC = C - A = [ - 2 , 2 , 0 ] = D - B = -→ BD Thus u = -→ AB = 1 ± -→ CD ² and v = -→ AC = 1 ± -→ BD ² . Hence

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x1a_sols_2 - Fall 2010 Math 251 Exam 1A Solutions c 2010...

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