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Chapter 13: 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
1. x虏 + 2y虏 = 8 2. z虏 + 2y虏 = 8 3. x虏 + 2z虏 = 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

Analyze the first cylinder equation

The equation of the first cylinder is given by \(x^2 + 2y^2 = 8\). Notice that the variable \(z\) is missing from this equation, which means the cylinder is parallel to the z-axis.
02

Analyze the second cylinder equation

The equation of the second cylinder is given by \(z^2 + 2y^2 = 8\). Notice that the variable \(x\) is missing from this equation, which means the cylinder is parallel to the x-axis.
03

Analyze the third cylinder equation

The equation of the third cylinder is given by \(x^2 + 2z^2 = 8\). Notice that the variable \(y\) is missing from this equation, which means the cylinder is parallel to the y-axis. So, the cylinders \(x^2 + 2y^2 = 8\), \(z^2 + 2y^2 = 8\), and \(x^2 + 2z^2 = 8\) are parallel to the z-axis, x-axis, and y-axis, respectively.

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