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To write the given equation for the hyperboloid of one sheet in standard form, let's start with the given equation:

\[
5x^2 + 2y^2 - 10z^2 - 10 = 0
\]

### Step-by-Step Simplification:

1. **Move the constant term to the other side**:
   \[
   5x^2 + 2y^2 - 10z^2 = 10
   \]

2. **Divide the entire equation by 10** to normalize the coefficients:
   \[
   \frac{5x^2}{10} + \frac{2y^2}{10} - \frac{10z^2}{10} = \frac{10}{10}
   \]

   Simplifying each term:
   \[
   \frac{x^2}{2} + \frac{y^2}{5} - z^2 = 1
   \]

### The standard form of the hyperboloid of one sheet is:
\[
\frac{x^2}{2} + \frac{y^2}{5} - z^2 = 1
\]

This is the required equation in standard form.

Would you like more details or have any further questions? Here are five related questions to deepen understanding:

1. What are the key characteristics of a hyperboloid of one sheet?
2. How does the standard form of a hyperboloid differ from that of an ellipsoid?
3. Can you describe the axis of symmetry for this hyperboloid?
4. How would the equation change if it described a hyperboloid of two sheets?
5. What would happen to the shape if the sign in front of \( z^2 \) were positive?

### Tip:
A hyperboloid of one sheet has two asymptotic directions and looks somewhat like a "saddle," curving inwards along one axis and outwards along the others.

Write the equation for the hyperboloid of one sheet given by 5x^2 + 2y^2 - 10z^2 - 10 = 0 in standard form.

Learn how to convert the equation 5x^2 + 2y^2 - 10z^2 - 10 = 0 into the standard form of a hyperboloid of one sheet. This solution includes detailed steps to simplify and normalize the equation, covering key concepts in analytic geometry and quadratic surfaces. Suitable for advanced high school or college-level students.

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