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Problem 562

Show how to find the semiaxes of the ellipse in which the plane $$ \begin{aligned} \mathrm{P}(\mathrm{x}, \mathrm{y}, z) &=\mathrm{pz}+\mathrm{qy}+\mathrm{rz} \\\ &=0(\mathrm{pqr} \neq 0) \end{aligned} $$ cuts the ellipsoid $E(x, y, z)=\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right)$ \(=1 .(0

Expert verified

To find the semiaxes of the ellipse formed by the intersection of the given plane and ellipsoid, first rewrite the plane equation to express z in terms of x and y: \(z = -\frac{q}{p}y - \frac{r}{p}x\). Then substitute this expression into the ellipsoid equation and simplify to obtain:
\(\frac{x^2(a^2p^2c^2+r^2p^2b^2c^2)}{a^2p^2c^2b^2} + \frac{y^2(a^2p^2c^2+q^2p^2b^2c^2)}{a^2p^2c^2b^2} - \frac{2qryx}{a^2p^2c^2} = 1\)
Let \(M = \sqrt{\frac{a^2p^2c^2+r^2p^2b^2c^2}{a^2p^2c^2b^2}}\) and \(N = \sqrt{\frac{a^2p^2c^2+q^2p^2b^2c^2}{a^2p^2c^2b^2}}\), the initial semiaxes of the ellipse before rotation. However, to find the actual semiaxes, you need to apply a rotation matrix and analyze the new equation of the ellipse.

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