prove that when you map a hyperbolic line $L$ from $\Bbb H^2$ to $\Bbb H$ that the result is a hyperbolic line $L'$ in $\Bbb H$

Question

I have that $T:(T,X,Y) \mapsto (\frac{-Y+i}{T-X})$ , with $(T,X,Y)\in \Bbb H^2$. I have to show that when I map a hyperbolic line $L$ from $\Bbb H^2$ to $\Bbb H$ that the result is a hyperbolic line $L'$ in $\Bbb H$

So I know that for $L$ $t^2=x^2+y^2+1$ and the equation for the plane through the origin is $At+Bx+Cy=0$

I'd like a hint, not a full answer, thank you in advance!