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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

Short Answer

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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Step 1: Analyzing the given equation of the plane and ellipsoid

We are given the equation of the plane P(x,y,z) as: \(pz + qy + rz = 0\), where pqr≠0. And the equation of the ellipsoid E(x,y,z) as: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) , where 0 < a < b < c. Our job is to find the equation of the intersection curve, which we know will be an ellipse, and then find its semiaxes.

Step 2: Solve for the ellipse equation formed by their intersection

To find the ellipse equation formed by their intersection, we need to express one variable from the plane equation in terms of the other two variables and substitute it into the ellipsoid equation. From the equation of the plane, we can rewrite it as: \(z = -\frac{q}{p}y - \frac{r}{p}x\) Substitute the above expression into the equation of the ellipsoid to get: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{(-\frac{q}{p}y - \frac{r}{p}x)^2}{c^2} = 1\) Simplify the equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{q^2 y^2 + 2qryx + r^2 x^2}{p^2 c^2} = 1\)

Step 3: Express the ellipse equation in the standard form to find the semiaxes

In this step, we will manipulate the equation we obtained in step 2 to express it in the standard form of an ellipse equation to get the semiaxes. An ellipse equation has the general form of: \(\frac{x^2}{S^2} + \frac{y^2}{T^2} = 1\) Where S and T are the semiaxes of the ellipse. Now, combine terms with x and y from the equation derived in Step 2: \(\frac{x^2(a^2p^2c^2+r^2p^2b^2c^2)}{a^2p^2c^2b^2} + \frac{y^2(a^2p^2c^2+q^2p^2b^2c^2)}{a^2p^2c^2b^2} - \frac{2qryx}{a^2p^2c^2} = 1\) Now, let's make substitution: Let \(M = \sqrt{\frac{a^2p^2c^2+r^2p^2b^2c^2}{a^2p^2c^2b^2}}\) represent x's coefficient and \(N = \sqrt{\frac{a^2p^2c^2+q^2p^2b^2c^2}{a^2p^2c^2b^2}}\) represent y's coefficient. Then we have: \( \frac{x^2}{M^2} + \frac{y^2}{N^2} - \frac{2qryx}{a^2p^2c^2} = 1 \) Please note that due to the subtraction term \(\frac{2qryx}{a^2p^2c^2}\), the ellipse is most likely to be rotated. So the analysis of the semiaxes can only be done after applying a rotation matrix and finding the new equation of the ellipse. However, based on the current scenario, we can state that M and N are the initial semiaxes of the ellipse before the rotation.

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