Hello, cazimi!
Here's what the Captain did . . .Identify the curve represented by: .y = cotθ + 1 .and .x = cot²θ
Rearrange the first equation: .cotθ .= .y - 1
Substitute into the second equation: .x .= .(y - 1)²
We have: .(y - 1)² .= .x
This is a parabola with vertex (0,1) which open to the right.