Hello, boardguy67!

I can help with one of your questions . . .

The set of equations defines a parametric curve in the xy-plane.

. .

My instructions are to find the lowest point on the curve.

Your point is correct, but labeled wrong on your graph.

It is at the very bottom of the loop, where there is a horizontal tangent.

Let be a value of the parameter.So while dinking about with this thing,

i became curious how i would find the point of intersection on the curve.

. . We have a point: .

Let be another value of the parameter .

. . We have another point: .

When will ?

We have: .

From [1]: .

. . Since , we have: . .[1]

From [2]: .

. . Since , we have: . .[2]

Substitute into [1]: .

. . .

Substitute into [2]: .

If

The curve intersects itself at .