Let's practise the basics, Part 3

Geometry Level 5

$68x^2+89y^2+104z^2-68xy+8xz-76yz=1296$

Find the volume $V$ of the solid region enclosed by the surface above. Enter $\frac{V}{\pi}$ .

The answer is 96.

1 solution

Otto Bretscher
Nov 22, 2015

The symmetric matrix $A$ of the quadratic form $q(x,y,z)=68x^2+89y^2+104z^2-68xy+8xz-76yz$ is $A=\begin{bmatrix}68&-34&4\\-34&89&-38\\4&-38&104\end{bmatrix}$ If $\lambda_k$ , for $k=1,2,3$ , are the eigenvalues of $A$ , then the semi-axes of the ellipsoid $q(x,y,z)=1296$ are $c_k=\sqrt{\frac{1296}{\lambda_k}}$ and the volume of the ellipsoid is $V=\frac{4\pi}{3}c_1c_2c_3=\frac{4\pi}{3}\sqrt{\frac{1296^3}{\det A}}=96\pi$ so that $\frac{V}{\pi}=\boxed{96}$

Are $c_k = \sqrt{\frac{\text{Given value}}{\lambda_k}}$ and $V = \frac{4\pi}3 c_1 c_2 c_3$ well known formulas? If so, can you provide/link me the proof?

Pi Han Goh - 5 years, 6 months ago

After a change of variables, the equation becomes

$\lambda_1u^2+\lambda_2v^2+\lambda_3w^2=1296.$ Now let $v=w=0$ to find the semi-axis for $u$ , etc.

To find the volume of the ellipsoid, start with the unit sphere, with volume $\frac{4\pi}{3}$ , and scale the axes by the various $c_k$ 's.

Otto Bretscher - 5 years, 6 months ago

Let's practise the basics, Part 3

The answer is 96.

1 solution

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