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I've been asked to find, if it exists, the minimum and maximum distance between the ellipsoid of ecuation $x^2+y^2+2z^2=6$ and the point $P=(4,2,0)$

To start, I've tried to find the points of the ellipsoid where the distance is minimum and maximum using the method of Lagrange multipliers.

First, I considered the sphere centered on the point $P$ as if it were a level surface and I construct:

$f(x,y,z)=(x-4)^2+(y-2)^2+z^2-a^2$

And I did the same with the ellipsoid, getting the following function:

$g(x,y,z)=x^2+y^2+2z^2-6$

Then, I tried to get the points where $\nabla$$f$ and $\nabla$$g$ are paralell:

$\nabla$$f=\lambda$$\nabla$$g$ $\rightarrow$ $(2x-8,2y-4,2z)=\lambda(2x,2y,4z)$

So, solving the system I got that $\lambda=1/2$, so in consquence $x=8$ and $y=4$

But when I replaced the values of $x,y$ obtained in $g=0$ to get the coordinate $z$ I get a complex number. I suspect there is something on my reasoning which is incorrect, can someone help me?

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Taking from where you left off. By equating coordinates of both sides, you have: $2z = 4\lambda z\implies z = 0$ or $\lambda = \dfrac{1}{2}$. If $\lambda = \dfrac{1}{2} \implies x = 8$ which is not possible since $x^2 \le 6$. So what is left is $z = 0$ or $\lambda = 0$. Also $\lambda \neq 1$ for otherwise $2x-8 = 2x$ which is not possible. $\lambda \neq 0$, for if it were $0$, then you have: $2x-8 = 0 \implies x = 4$, and this is not possible since $x^2 \le 6$. Thus $2x-8 = 2\lambda x, 2y-4 = 2\lambda y\implies x = \dfrac{4}{1-\lambda}, y = \dfrac{2}{1-\lambda}\implies \dfrac{16}{(1-\lambda)^2}+\dfrac{4}{(1-\lambda)^2}=6\implies (1-\lambda)^2 = \dfrac{20}{6} = \dfrac{10}{3}\implies \lambda = 1\pm \dfrac{\sqrt{30}}{3}$. Can you take it from here ?. It looks as if one of the lambdas will yield a min and the other a max.

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  • $\begingroup$ Following yor suggestions, I've got two points: $Q_{1}(x,y,z)=(-12/\sqrt{30}$$,-6/\sqrt{30},0)$ and $Q_{2}(x,y,z)=(12/\sqrt{30}$$,6/\sqrt{30},0)$. Now, from here to get the minimum and maximum distance I could do it through $|Q_{i}P|=\sqrt{(x-4)^2+(y-2)^2+z^2}$, right? $\endgroup$ – Neisy Sofía Vadori Jun 17 '17 at 16:12
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    $\begingroup$ Yep.Try it and let me know your answer. $\endgroup$ – DeepSea Jun 17 '17 at 21:47
  • $\begingroup$ I obtained that $|Q_{1}P|=\sqrt{36+10/3\sqrt{30}}$, and $|Q_{2}P|=\sqrt{36-10/3\sqrt{30}}$. What do you think @DeepSea? $\endgroup$ – Neisy Sofía Vadori Jun 19 '17 at 19:16
  • $\begingroup$ I think you got the right answer. Are you good at physics also? or so so? $\endgroup$ – DeepSea Jun 19 '17 at 23:27
  • $\begingroup$ haha I think I am, but I'm in my first year of the university, so in process of getting way much better! $\endgroup$ – Neisy Sofía Vadori Jun 20 '17 at 13:56
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Since by C-S $$2x+y\leq\sqrt{(x^2+y^2)(2^2+1^2)}=\sqrt{5(x^2+y^2)},$$ we obtain $$(x-4)^2+(y-2)^2+z^2=x^2+y^2+z^2-4(2x+y)+20\geq $$ $$\geq x^2+y^2+z^2-4\sqrt{5(x^2+y^2)}+20=6-2z^2+z^2-4\sqrt{5(6-2z^2)}+20=$$ $$=26-z^2-4\sqrt{5(6-2z^2)}\geq26-4\sqrt{30}.$$ It's obvious that the equality occurs, which gives the minimal value: $$\sqrt{26-4\sqrt{30}}.$$

Since $-2x-y\leq\sqrt{5(x^2+y^2)}$, by the same way we can get a maximal value: $$\sqrt{26+4\sqrt{30}}.$$

Done!

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hint The ellipse can be parametrized as

$$x_e =\sqrt {6}\sin (\phi)\cos (\theta) $$ $$y_e=\sqrt{6}\sin (\phi)\sin (\theta ) $$ $$z_e=\sqrt {3}\cos (\phi) $$

the square of the distance from a point of the ellipse to the point $(4,2,0) $ is

$$D^2=(x_e-4)^2+(y_e-2)^2+z_e^2$$ $$=3\sin^2 (\phi)+23-8x_e-4y_e $$

You can find min and max D,

by solving the system, $$\frac {\partial D^2}{\partial \phi}=\frac {\partial D^2}{\partial \theta}=0$$

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  • $\begingroup$ In my course, I didn't learn yet how to parametrize through polar coordinates. In which book can I read something about it? $\endgroup$ – Neisy Sofía Vadori Jun 17 '17 at 16:18
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    $\begingroup$ @NeisySofíaVadori Look for spherical coordinates. $\endgroup$ – hamam_Abdallah Jun 17 '17 at 16:54

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