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237t2s - 1 Let the surface S be parametrized by f(u v = 2 u...

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1. Let the surface S be parametrized by ) , , 2 ( ) , ( 2 v u e u v u v u v + + + = f and let be a point in ) 1 , 2 , 2 ( 0 P 3 R . (a) [2 marks] Show that the point lies on the surface S . 0 P We should be able to find ( u , v ) such that ) 1 , 2 , 2 ( ) , ( = v u f . We have the system of equations 1 , 2 , 2 2 2 = + = + = + v u e u v u v . From the second equation we get since on the right hand side we do not have e and also u =1, to satisfy that equation. 0 = v Its easy to see that the other two equations are satisfied also. So lies on the surface S 0 P and it corresponds to ) 0 , 1 ( ) , ( = v u (b) [5 marks] Write the equation for the tangent plane to S at the point in the 0 P form d cz by ax = + + , where are constants. d c b a , , , Vector normal to the surface S at the point is 0 P ) 0 , 1 )( ( v u × = f f N , ) 2 , 1 , 1 ( ) 2 , 1 , 1 ( ) 0 , 1 )( ( 0 1 = + = = = v u u v u u f , ) 1 , 1 , 1 ( ) 1 , , 1 ( ) 0 , 1 )( ( 0 1 = + = = = v u v e v u v f . So N = = ) 1 , 1 , 1 ( ) 2 , 1 , 1 ( × ) 2 , 1 , 3 ( and the equation of the tangent plane to S at the point is 0 P 0 ) 1 )( 2 ( ) 2 ( 1 ) 2 ( 3 = + + z y x or 6 2 3 = + z y x . 2
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