Find the vector equation that represents the curve of intersection of the paraboloid z=3x^2+y^2 and the surface x=y^5. Write the equation so that one of the functions is simply t.

Answer 1

We note that z(t) is a function of x(t) and y(t) and that x(t) is a function of y(t), so we will parametrize y(t) to simply be t.

From here, we note that x(t) = y⁵ = t⁵.

Lastly, z(t) = 3x² + y² = 3(t⁵)² + t² = 3t¹⁰ + t²

So we have <t⁵, t, 3t¹⁰ + t²> is our desired vector!