To find the parametric equations for the intersection of the given surfaces, we first need to express the equations in a way that allows us to introduce a parameter. Here are the two surfaces:

1. The paraboloid: $z = 2x^2 + y^2$
2. The parabolic cylinder: $y = 2x^2$

Step-by-Step Solution

1. Express $y$ in terms of $x$:

   From the equation of the parabolic cylinder: $y = 2x^2$

2. Substitute $y$ into the paraboloid equation:

   Substitute $y = 2x^2$ into the equation for the paraboloid: $z = 2x^2 + (2x^2)^2$

3. Simplify the equation for $z$:

   $z = 2x^2 + 4x^4$

4. Introduce a parameter:

   Let's introduce a parameter $t$ such that $x(t) = t$. Then, using the previous expressions: $y(t) = 2x(t)^2 = 2t^2$ $z(t) = 2x(t)^2 + 4x(t)^4 = 2t^2 + 4t^4$

Parametric Equations

Now, we can write the parametric equations of the curve:

$x(t) = t$ $y(t) = 2t^2$ $z(t) = 2t^2 + 4t^4$

These equations describe the intersection of the paraboloid and the parabolic cylinder in terms of the parameter $t$.

Would you like a further explanation or have more questions?

Related Questions

1. How would the parametric equations change if the cylinder were different, say $y = 3x^2$?
2. What is the geometric interpretation of the curve formed by this intersection?
3. How can we verify that these parametric equations satisfy both the original surface equations?
4. What are the limits of $t$ for which the parametric equations are valid?
5. How would the parametric equations be modified if the paraboloid were $z = 3x^2 + y^2$?

Tip

When finding the intersection of surfaces, it often helps to express one of the variables in terms of a parameter and substitute into the other equations to simplify the process.