Describe the following surface and give traces if available.
x^(2)-4z^(2)=4
just a regular old hyperbola, in the x-z plane.
Please send me the solution
The given equation x^2 - 4z^2 = 4 represents a surface in three-dimensional space. To describe the surface, let's manipulate the equation and analyze it further.
First, let's rewrite the given equation in terms of z:
4z^2 = x^2 - 4
By rearranging the terms:
z^2 = (x^2 - 4)/4
z^2 = (1/4)x^2 - 1
Now, we notice that z^2 is isolated. This indicates that the surface is a double cone, also known as a hyperboloid of two sheets.
To find the traces of this surface, we need to examine the equation by fixing one variable and allowing the others to vary.
1. x-axis trace (yz-plane): Set x = 0 in the equation:
0^2 - 4z^2 = 4
-4z^2 = 4
z^2 = -1
Since the square of a real number cannot be negative, there is no x-axis trace. This means that the hyperboloid does not intersect the yz-plane.
2. y-axis trace (xz-plane): Set y = 0 in the equation:
x^2 - 4z^2 = 4
x^2 - 4z^2 = 4
Simplifying the equation, we obtain:
x^2 = 4 + 4z^2
From this equation, we can see that the y-axis trace is an ellipse centered at x = 0 with a positive semi-major axis equal to √(4 + 4z^2).
3. z-axis trace (xy-plane): Set z = 0 in the equation:
x^2 - 4(0^2) = 4
x^2 = 4
x = ±2
For the z-axis trace, we have two points, (2, 0, 0) and (-2, 0, 0), which lie on the surface.
To summarize:
- The surface described by the equation x^2 - 4z^2 = 4 is a hyperboloid of two sheets.
- It does not intersect the yz-plane (no x-axis trace).
- The y-axis trace is an ellipse centered at x = 0.
- The z-axis trace consists of two distinct points: (2, 0, 0) and (-2, 0, 0).