(x-2)^2-y^2-(z+1)^2=0 What surface is this?

The equation given is:

\[
(x - 2)^2 - y^2 - (z + 1)^2 = 0
\]

This represents a **hyperbolic paraboloid**. Let’s break it down step by step.

### Step 1: Rearrange the equation
We can rewrite it as:

\[
(x - 2)^2 = y^2 + (z + 1)^2
\]

This form is useful for understanding the surface. It suggests that the value of \((x - 2)^2\) is always the sum of squares of \(y\) and \(z + 1\), which hints at the structure of the surface.

### Step 2: Identify the standard form
The equation resembles the general form of a **hyperbolic paraboloid**:

\[
x^2 = y^2 + z^2
\]

but shifted in space:
- Shifted by 2 units along the x-axis (\(x - 2\))
- Shifted by -1 unit along the z-axis (\(z + 1\))

### Step 3: Interpretation
This surface is a **hyperbolic paraboloid** that opens along the x-axis, with its center shifted to \((2, 0, -1)\).

Would you like further details on hyperbolic paraboloids, or have any specific questions?

### Related Questions:
1. How does a hyperbolic paraboloid differ from an elliptical paraboloid?
2. How can you rotate this surface along an axis?
3. What are the cross-sectional shapes of this surface in the yz-plane?
4. What are the key features of hyperbolic paraboloids in real-world applications?
5. How would this equation change if there were additional coefficients?

### Tip:
To better visualize surfaces, it's helpful to sketch their cross-sections in different planes (xy, xz, yz).

(x-2)^2 - y^2 - (z+1)^2 = 0 What surface is this?

Explore how the equation (x-2)^2 - y^2 - (z+1)^2 = 0 represents a hyperbolic paraboloid. Learn the step-by-step process to identify this quadratic surface and understand the geometric shifts in space. Suitable for advanced high school or college students studying geometry or quadratic surfaces.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation