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Chapter 12: Problem 5

To which coordinate axes are the following cylinders in \(\mathbb{R}^{3}\) parallel: \(x^{2}+2 y^{2}=8, z^{2}+2 y^{2}=8,\) and \(x^{2}+2 z^{2}=8 ?\)

Short Answer

Expert verified
Cylinder 1: \(x^{2}+2y^{2}=8\) Cylinder 2: \(z^{2}+2y^{2}=8\) Cylinder 3: \(x^{2}+2z^{2}=8\) Answer: The first cylinder is parallel to the z-axis, the second cylinder is parallel to the x-axis, and the third cylinder is parallel to the y-axis.

Step by step solution

01

Determine the Variables

In this equation, we have \(x^{2}\) and \(2y^{2}\) terms. Since there is no \(z\) component, this cylinder must be parallel to the \(z\)-axis. For the second cylinder: \(z^{2}+2y^{2}=8\).
02

Determine the Variables

In this equation, we have \(z^{2}\) and \(2y^{2}\) terms. Since there is no \(x\) component, this cylinder must be parallel to the \(x\)-axis. For the third cylinder: \(x^{2}+2z^{2}=8\).
03

Determine the Variables

In this equation, we have \(x^{2}\) and \(2z^{2}\) terms. Since there is no \(y\) component, this cylinder must be parallel to the \(y\)-axis. To summarize: - The cylinder defined by \(x^{2}+2y^{2}=8\) is parallel to the \(z\)-axis. - The cylinder defined by \(z^{2}+2y^{2}=8\) is parallel to the \(x\)-axis. - The cylinder defined by \(x^{2}+2z^{2}=8\) is parallel to the \(y\)-axis.

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