Minimal distance between the origin and intersection of $x^2 = 2yz$ and $x^2+3y^2+2z^2 = 30$

Question

Find the minimal distance between the origin and intersection of $x^2 = 2yz$ and $x^2+3y^2+2z^2 = 30$

Attempt: $2yz+3y^2+2z^2 = 30$

I called $f(y,z) = 2yz+3y^2+2z^2$

Then, $\nabla f(y,z) = (2z+6y,2y+4z) = 0 \Rightarrow y=z=0$

So the critical point is $(0,0,0)$, which is clearly a minimum. But, I think this is wrong, because the minimum point is not necessarily the minimal distance to the origin. How do I get the closest point to the origin?

Answer 1

I'll try to answer my own question based on the hints.

Define $f(x,y,z)=x^2+y^2+z^2$, $\varphi_1(x,y,z)=x^2-2yz$ and $\varphi_2(x,y,z)=x^2+3y^2+2z^2$

By doing $\nabla f = \lambda_1\varphi_1+\lambda_2\varphi_2$, we have

$(2x,2y,2z)=\lambda_1(2x,-2z,-2y)+\lambda_2(2x,6y,4z)$

$2x = \lambda_12x+\lambda_22x$

$2y =-\lambda_12z+\lambda_26y$

$2z=-\lambda_12y+\lambda_24z$

Together with $\varphi_1 = 0$ and $\varphi_2=30$, calculating this system of equation, we have:

$y=(3/2)z$ and $x =\pm\sqrt3z$, so $z=\pm\sqrt{60/19}$

And the points are $(\pm\sqrt3\sqrt{60/19},(3/2)\sqrt{60/19},\sqrt{60/19})$ and $(\pm\sqrt3\sqrt{60/19},-(3/2)\sqrt{60/19},-\sqrt{60/19})$

which are all minimum points as $\det Hf(x,y,z) = 8>0$, as

$Hf(x,y,z)=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}$

Answer 2

HINT

The minimum distance to the origin is given by the Euclidean norm. To make the problem less complex, we can solve the equivalent problem, minimizing the square of the Euclidean distance $D(x,y,z) = x^2 + y^2 + z^2$ subject to both of your equality constraints, $x^2 = 2yz$ and $x^2+3y^2+ 2z^2 = 30$.

You don't have to solve it using Lagrange multipliers necessarily, but that indeed would be a very elegant way of doing so.

Answer 3

i have $$\sqrt{x^2+y^2+z^2}=f(x,y,z)$$ for $$x^2$$ we have $$x^2=2yz$$ and further $$2z^2=30-x^2-3y^2=2yz-3/2y^2$$ we get $$z^2=15-yz-\frac{3}{2}y^2$$ and we have to optimize $$\sqrt{yz-\frac{1}{2}y^2+15}$$ you can consider the function $$h(y,z)=yz+15-\frac{1}{2}y^2$$ and compute the partial derivatives with respect to $$y,z$$