I am having trouble with a problem that I found in a multivariate calculus book. The problem is :
Let $a$, $b$, and $c$ be nonzero real numbers. Find the minimum volume bounded by the planes given by $x=0$, $y=0$, $z=0$, and a plane that is tangent to the ellipsoid given
by $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$.
I suspect that solving this problem would involve the use of the Lagrange multiplier method.
In my solution so far I start with defining a function :
\begin{equation}
h(x,y,z) = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} – 1
\end{equation}
So that the ellipse (call it $E$) is given by :
\begin{equation}
E = \{ (x,y,z) \in \mathbb{R}^{3} \; : \; h(x,y,z) = 0 \}
\end{equation}
We have the partial derivatives of $h$ as :
\begin{align}
h_{x}(x,y,z) & = \frac{2x}{a^{2}}\\
h_{y}(x,y,z) & = \frac{2y}{b^{2}}\\
h_{z}(x,y,z) & = \frac{2z}{c^{2}}
\end{align}
Now let $(x_{0},y_{0},z_{0}) \in E$. The tangent plane to E at $(x_{0},y_{0},z_{0})$ can be defined implicitly using a function $T$ :
\begin{align}
T_{x_{0},y_{0},z_{0}}(x,y,z) & = h_{x}(x_{0},y_{0},z_{0})(x-x_{0}) + h_{y}(x_{0},y_{0},z_{0})(y-y_{0}) + h_{z}(x_{0},y_{0},z_{0})(z-z_{0}) \\
& = \frac{2x_{0}(x-x_{0})}{a^{2}} + \frac{2y_{0}(y-y_{0})}{b^{2}} + \frac{2z_{0}(z-z_{0})}{c^{2}}\\
\end{align}
Where the tangent plane (say $Q(x_{0},y_{0},z_{0})$) is :
\begin{equation}
Q(x_{0},y_{0},z_{0}) = \{(x,y,z) \in \mathbb{R}^{3} \; : \; T_{x_{0},y_{0},z_{0}}(x,y,z) =0 \}
\end{equation}
So I suspect the point of the problem is to find the $(x_{0},y_{0},z_{0}) \in E$ so that
the volume bounded by the planes defined by $Q(x_{0},y_{0},z_{0})$ , $x = 0$, $y=0$, and $z = 0$ is minimized. The problem that I have at this point is how to get an expression for this volume.
Can anyone help with this ?
Best Answer
It is much, much easier to perform a coordinate transformation that takes the ellipse to the sphere: $$(u, v, w) = (x/a, y/b, z/c)$$ and then observe that this transformation preserves tangency and scales volume proportionally by a factor of $1/(abc)$.
Then, since the sphere is symmetric, the minimum volume in the $(u,v,w)$ coordinate space is attained when the point of tangency is $u = v = w = 1/\sqrt{3}$ (why?). Solve for the points where this plane intersects the $u$-, $v$-, and $w$-axes, compute the volume of the tetrahedron, and reverse the scaling transformation by multiplying by $abc$.