Question

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. the paraboloid z = 7x2 + y2 and the parabolic cylinder y = 3x2

Answer 1

The vector function r(t) representing the curve of intersection of the surfaces z = 7x² + y² and y = 3x² is r(t) = it + 3t²j + (7t² + 9t´)k.

Step-by-step explanation:

Finding a Vector Function for a Curve of Intersection

To find the vector function r(t) that represents the curve of intersection of the surfaces z = 7x² + y² and y = 3x², we can parameterize the variables using a suitable parameter, commonly denoted as t. We can set x = t, which imminently provides y as y = 3t² by substituting t into the equation of the parabolic cylinder. Then, substituting both x and y into the equation of the paraboloid gives us z = 7t² + (3t²)² = 7t² + 9t´. Therefore, the vector function is:

r(t) = it + j3t² + k(7t² + 9t´).

This function r(t) represents all points (x, y, z) on the curve where the two surfaces intersect. Since the curve lies on both surfaces simultaneously, z must be equal for both equations when x and y from the curve are substituted into them.

Answer 2

Letting
`x=t`, we get
`y=3x^2=3t^2` and
`z=7x^2+y^2=7t^2+(3t^2)^2=7t^2+9t^4`, so we can parameterize the intersection by


`\mathbf r(t)=\langle t,3t^2,7t^2+9t^4\rangle`

where
`-\infty<t<\infty`.

Image attached.