Answer
$(2,1,\sqrt {5}), (2,1,-\sqrt {5})$
Work Step by Step
Given: Equation of cone: $z^2=x^2+y^2$ ; point $(4,2,0)$
Need to apply Lagrange Multipliers Method to determine the shortest distance from the point $(4,2,0)$ to the plane $z^2=x^2+y^2$
we have $\nabla f=\lambda \nabla g$
Rewrite the equation of cone as: $z=\sqrt {x^2+y^2}$
The closet distance can be found as:
$d=\sqrt{x-l)^2+(y-m)^2+(z-n)^2}$
or, $d=\sqrt{(x-4)^2+(y-2)^2+(x^2+y^2)}$
$f(x,y)=d^2=(x-4)^2+(y-2)^2+(x^2+y^2)$
Also, $f_x=4x-8, f_y=4y-4$
Simplify to get the values for $x$ and $y$.
we have $x=2$ and $y=1$
Therefore, $z=(2)^2+(1)^2=\pm \sqrt {5}$
Hence, our result is: $(2,1,\sqrt {5}), (2,1,-\sqrt {5})$
