Problem 10

Question

What is the name of the surface defined by the equation $-y^{2}-\frac{z^{2}}{2}+x^{2}=1 ?$

Step-by-Step Solution

Verified

Answer

Question: Identify the surface defined by the equation $-y^2-\frac{z^2}{2}+x^2=1$. Answer: The surface is a Hyperboloid of One Sheet.

1Step 1: Rewrite the given equation in standard form

First, let's rewrite the equation as: $x^{2} - y^{2} - \frac{z^{2}}{2} = 1$

2Step 2: Identify the type of surface

The given equation is similar to the standard form of a hyperboloid of one sheet which has the formula: $\frac{x^{2}}{a^2} - \frac{y^{2}}{b^2} - \frac{z^{2}}{c^2} = 1$ Comparing the given equation with the standard form, we can see that $a^2 = 1$, $b^2 = 1$, and $c^2 = 2$. Therefore, the equation represents a hyperboloid of one sheet.

3Step 3: Provide the final answer

The surface defined by the equation $-y^{2}-\frac{z^{2}}{2}+x^{2}=1$ is called a Hyperboloid of One Sheet.

Problem 10

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