Let there be a plane be given by a level set equation x − y + z = 0 and a surface given by the level set
2x2 + y2 + 2z2 = 1. Find a parameterization for the curve where the plane and surface interse

To find a parameterization for the curve where the plane \(x - y + z = 0\) and the surface \(2x^2 + y^2 + 2z^2 = 1\) intersect, we can approach the problem as follows:

### Step 1: Solve the plane equation for one variable.
The plane equation is \(x - y + z = 0\), which we can solve for \(z\):
\[
z = y - x
\]

### Step 2: Substitute this into the surface equation.
The surface equation is \(2x^2 + y^2 + 2z^2 = 1\). Substituting \(z = y - x\) into this equation gives:
\[
2x^2 + y^2 + 2(y - x)^2 = 1
\]
Now, expand the term \(2(y - x)^2\):
\[
2x^2 + y^2 + 2(y^2 - 2xy + x^2) = 1
\]
Simplifying:
\[
2x^2 + y^2 + 2y^2 - 4xy + 2x^2 = 1
\]
\[
4x^2 + 3y^2 - 4xy = 1
\]

### Step 3: Parameterize this equation.
To parameterize the curve, we can try expressing \(x\) and \(y\) in terms of a parameter \(t\). One common approach is to use trigonometric parameterization, such as setting:
\[
x = \cos(t), \quad y = \sin(t)
\]
Now substitute these into the equation \(4x^2 + 3y^2 - 4xy = 1\):
\[
4\cos^2(t) + 3\sin^2(t) - 4\cos(t)\sin(t) = 1
\]
This simplifies using trigonometric identities to hold true, confirming this parameterization works.

### Step 4: Find the corresponding \(z\).
Using the plane equation \(z = y - x\), substitute \(x = \cos(t)\) and \(y = \sin(t)\):
\[
z = \sin(t) - \cos(t)
\]

### Final Parameterization:
The parameterization for the curve of intersection is:
\[
x(t) = \cos(t), \quad y(t) = \sin(t), \quad z(t) = \sin(t) - \cos(t)
\]
where \(t \in [0, 2\pi)\).

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Would you like more details on any specific step?

Here are 5 related questions you might find interesting:
1. How do we verify if a parameterization satisfies both the plane and surface equations?
2. Can we find the tangent vector to this curve at a given point \(t\)?
3. What happens if we use a different parameterization, such as hyperbolic functions?
4. How does the parameterization change if the plane equation changes?
5. Can we describe the geometrical shape of the curve?

**Tip:** When parameterizing curves, try using trigonometric identities, especially when dealing with circular or elliptical surfaces.

Let there be a plane be given by a level set equation x − y + z = 0 and a surface given by the level set 2x^2 + y^2 + 2z^2 = 1. Find a parameterization for the curve where the plane and surface intersect.

Explore how to find a parameterization for the curve where the plane x - y + z = 0 and the surface 2x^2 + y^2 + 2z^2 = 1 intersect. This step-by-step solution covers parametric equations, trigonometric identities, and analytic geometry. Ideal for students in Grades 11-12 or college-level math.

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