Find the minimum distance from the point 12131213 to the par

Find the minimum distance from the point (1,2,13)(1,2,13) to the paraboloid given by the equation z=x2+y2z=x2+y2.

Minimum distance =

D = sqrt [(x-1)^2 +(y-2)^2 +(z-13)^2]

we need to minimize D

f(x,y,z) = [(x-1)^2 +(y-2)^2 +(z-13)^2]

f\'(x) = 2(x-1)

f\'(y) = 2(y-2)

f\'(z) = 2(z-13)

Constraint given are

g(x,y,z) = x^2 + y^2 - z

Equation will be

x^2 + y^2 - z = 0

and from partial derivatives

df = dg

2(x-1) = 2*a*x

2(y-2) = 2ay

2(z-13) = -a

from these equations:

x = 1/(1 - a), and y = 2/(1 - a) and z = 13 - a/2

substituting in first equation

x^2 +y^2 - z = 0

1/(1-a)^2 + 4/(1 - a)^2 - (13 - a/2) = 0

a^3 - 28a^2 + 53a - 16 = 0

Solving this equation

a1 = 1.64 or a2 = 25.98 or a3 = 0.375

from point a1

(x, y, z) = (-1.56, -3.125, 12.18)

from point a2

(x, y, z) = (-0.04, -0.08, 0.01)

from point a3

(x, y, z) = (1.6, 3.2, 12.81)

Calculating the distance from these points,

we can see that minimum distance will be from point a3

D = sqrt [(1 - 1.6)^2 + (2 - 3.2)^2 + (13 - 12.81)^2]

D = 1.355