Question

Find the minimum distance between the origin and the

surface x^2y - z^2 + 9 = 0.

Answer 1

Given: Equation x²y - z² + 9 = 0.

To find:  The minimum distance between the origin and the surface.

Solution:

• Now, lets consider a point P(x,y,z) on the surface x²y - z² + 9 = 0.

• Formula for the distance from the origin is

                 d = √{x²+y²+z²}

• Let f(x,y,z) = x²+y²+z² and g(x,y,z) =  x²y - z² + 9

              x²y - z² + 9 = 0

              z² = x²y + 9

• Putting this value in f(x,y,z), we get:

             f(x,y,z) = x²+y²+x²y + 9

• Now we have got the function in two variables.

• Assume f to be (x,y) = x²+y²+x²y + 9

• Now,

            ∂x /∂F  = 0 ;  2x+2xy=0

            and   ∂y /∂F  = 0 ; 2y+x²=0

• After solving the 1st equation, we get, x=0  or  y=−1.

• After solving the 2nd equation, we get x=y=0 or x=±√2 and y=-1.

• When x=y=0,

              z² = 9, z = ±3

• When x=±√2 or y=-1,

              z² = -2 + 9 = 7, z² = ±√7

• Now,  the critical points will be: (0,0,±3) and (±√2,-1,±√7)

• Now,

              f( 0 , 0 , ±3 ) = 9

             f(±√ 2 , −1, ±√ 7  ) = 10

Answer:

       So the minimum distance is 3 which is at the point  f( 0 , 0 , ±3 ).

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