251sol3 - 5 3 y p 5 3 P 2 3-3 5 3 2(6 = 4 x 2 4 z 2 3 p y 5...

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Assignment 3 Multivariate Calculus Math 251 (Fall 2010) A1. A surface consists of all points P such that the distance from P to the plane y = 1 is twice the distance from P to the point (0 , - 1 , 0). Find an equation for the surface and identify it. Solution Let P ( x, y, z ) be a point in R 3 . Let f ( x, y, z ) be the distance from ( x, y, z ) to the plane y = 1. Let g ( x, y, z ) be the distance from ( x, y, z ) to the point (0 , - 1 , 0). Then f ( x, y, z ) = | y - 1 | and g ( x, y, z ) = r ( x - 0) 2 + ( y - ( - 1)) 2 + ( z - 0) 2 = r x 2 + ( y + 1) 2 + z 2 . We are asked to solve the equation f ( x, y, z ) = 2 g ( x, y, z ). | y - 1 | = 2 r x 2 + ( y + 1) 2 + z 2 (1) ( y - 1) 2 = 4( x 2 + z 2 + ( y + 1) 2 ) (2) y 2 - 2 y + 1 = 4 x 2 + 4 z 2 + 4 y 2 + 8 y + 4 (3) 0 = 4 x 2 + 4 z 2 + 3 y 2 + 10 y + 3 . (4) We need to complete the square on the last three terms. 0 = 4 x 2 + 4 z 2 + 3( y 2 + 2 · 5 3 y ) + 3 (5) = 4 x 2 + 4 z 2 + 3( y 2 + 2 ·
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Unformatted text preview: 5 3 y + p 5 3 P 2 ) + 3-3( 5 3 ) 2 (6) = 4 x 2 + 4 z 2 + 3 p y + 5 3 P 2 + 3-3 p 5 3 P 2 (7) = 4 x 2 + 4 z 2 + 3 p y + 5 3 P 2-16 3 . (8) We can already see that this is an elipsoid centred at (0 ,-5 3 , 0). Optionaly, we can farther manipulate this to look like one of the standard forms on P. 808 of the text. 4 x 2 + 4 z 2 + 3 p y + 5 3 P 2 = 16 3 (9) 3 16 (4 x 2 + 4 z 2 + 3 p y + 5 3 P 2 = 1 (10) 3 4 x 2 + 3 4 z 2 + 9 16 p y + 5 3 P 2 = 1 (11) x 2 ± r 4 / 3 ² 2 + ( y--5 3 ) 2 (4 / 3) 2 + z 2 ± r 4 / 3 ² 2 = 1 . (12) The eliploid is obtained from a sphere by “stretching” it by a factor of r 4 / 3 in the y direction. a...
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This note was uploaded on 12/13/2010 for the course MATH 251 taught by Professor Unknown during the Spring '08 term at Simon Fraser.

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251sol3 - 5 3 y p 5 3 P 2 3-3 5 3 2(6 = 4 x 2 4 z 2 3 p y 5...

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