Problem 5
Question
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 \(x^{2}+2 z^{2}=8 ?\)
Step-by-Step Solution
Verified Answer
1. x² + 2y² = 8
2. z² + 2y² = 8
3. x² + 2z² = 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.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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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