I am having trouble with the following multivariabel calculus problem regarding gradients and surfaces:
I am told to find all tangent planes to the ellipsoid: $x^2+2y^2+3z^2=20$, which are parallel to the plane given by $v_1=[0,5,0]$ and $v_2=[1,5,-1]$. I get that by the cross product I can find the normal to the plane in question and thus find that $n=[5,0,5]$.
However, here I am having issues continuing, I want to say that $\nabla f(x,y,z)=[2x,4y,6z]$ which is a normal to the surface at the point $(x,y,z)$ and thus get that the normal to the tanget plane is $n_{tangent}=[5/2,0,5/6]$ but this is completely wrong, since according to the keys, there should be two of these tanget planes, and my result would only yield one.
So my question is really how to set this problem up, I cannot see how to find more than one solution, a hint or some help would be greatly appreciated.
Best Answer
This is a very standard Lagrange multiplier problem. $$\mathscr{L}(x,y,z,\lambda)=f(x)-\lambda g(x))$$ Here the function to be optimized is the plane, $f(x)=x+z$ and the constraint is the ellipsoid.
$$\mathscr{L}=x+z-\lambda\left(x^2+2y^2+3z^2-20\right)$$ $$\nabla\mathscr{L}=\left(\begin{array}{c}-2\;\lambda\;x+1\\-4\;\lambda\;y\\-6\;\lambda\;z+1\\-x^{2}-2\;y^{2}-3\;z^{2}+20\end{array}\right)=\left(\begin{array}{c}0\\0\\0\\0\end{array}\right)$$ The solution to these:$$ \left\{ \left\{ x = -\sqrt{15}, y = 0, z = -\frac{\sqrt{15}}{3}, \lambda = -\frac{\sqrt{15}}{30} \right\} , \left\{ x = \sqrt{15}, y = 0, z = \frac{\sqrt{15}}{3}, \lambda = \frac{\sqrt{15}}{30} \right\} \right\} $$ and of course you don't care about $\lambda$. One of the planes is $$\left(\begin{array}{c}5\\0\\5\end{array}\right)\cdot\left(\begin{array}{c}x\\y\\z\end{array}\right)=\left(\begin{array}{c}-\sqrt{15}\\0\\-\frac{\sqrt{15}}{3}\end{array}\right)\left(\begin{array}{c}5\\0\\5\end{array}\right)$$ and the other one is with the two negative values changed to positive.