Q: how to find path in complex plane? Could anyone help me how to find a path in $\mathbb{C}$ plane
(1) with initial point $1-i$ and terminal point $-2+4i$ having arc length $100$ unit?
(2) How to find a smooth figure 8 path in plane?
(3) Find a path that traces $e^{2\pi it}+1$ in the opposite direction and 3 times faster.
I have simply no idea how to proceed. for the first one I though about $xt+(1-t)y$

A: (1)
Let $1-i\to a-i\to a+4i\to-2+4i$ and find $a\in\mathbb{R}$.
(2)
Probably you mean Figure Eight Curve
(3)
$z=e^{2\pi it}+1$ is the circle $|z-1|=1$ or $z=1+\cos2\pi t+i\sin2\pi t$ where $0\leq t\leq1$. In this case with $1-t$ the path traced in apposite direction and the factor $\dfrac{t}{3}$ instead of $t$ caused the path spanned 3 times faster. So desired path is $z=1+\cos2\pi(1-\dfrac{t}{3})+i\sin2\pi(1-\dfrac{t}{3})$ where $0\leq t\leq3$.

A: Hint for (1) assuming an arc of circle :
The center $O$ of a circle that passes through $A=(1,-1)$ and $B=(-2,4)$ is located on the perpendicular bissector of $[AB]$ and the arc length is $l=R\theta$ or $l=R(2\pi-\theta)$ where $\theta=\widehat{AOB}$
