Hello, paronga!

Your idea is a good one!

So i have the curve: .

And i need to show that it has one self-intersection point.

So far, i've thought maybe if : .

where: . . Yes!

So we have: .

From [1], we have: .

. .

We have: .

. . . . .

From [2], we have: .

. .

. .

. .

Substitute [3] into [4]: .

. . which simplifies to: .

. . . . . . . . . . . . . . .

It turns out that: .

Therefore, there isself-intersection point.one