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Hyperbolic cylinder: the desired figure and a figure obtained from code
I what to get a Hyperbolic cylinder $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

How to modify the code: Please help.
\documentclass[border=2mm]{standalone}
\usepackage    {tikz}
\usetikzlibrary{3d}

\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,x={(-0.5cm,-0.6cm)},y={(0.9333cm,-0.1646cm)},z={(0cm,0.5cm)}]
  % coordinates
  \coordinate (A1) at ({-4+2*cos(60)}, 1.75,{ 2*sin(60)});
  \coordinate (C1) at ({-4+2*cos(60)},-1.75,{-2*sin(60)});
  \coordinate (A2) at ({ 4-2*cos(60)}, 1.75,{ 2*sin(60)});
  \coordinate (C2) at ({ 4-2*cos(60)},-1.75,{-2*sin(60)});
  % cylinder back
  \begin{scope}[canvas is xz plane at y=-1.75]
    \shade [opacity=0.7,left color=blue!10,right color=blue!70,shading angle=120]
      (A1) arc (60:-60:2) -- (C1) arc (-60:60:2) -- cycle;
  \end{scope}
   % y,z axis
  \draw [-latex] (0,0,0) -- (0,3,0) node [right] {$y$};
  \draw [-latex] (0,0,0) -- (0,0,3) node [above] {$z$};
  % front cylinder 
  \begin{scope}[canvas is xz plane at y=-1.75]
    \shade [opacity=0.7,left color=gray!10,right color=gray!70,shading angle=30]
      (A2) arc (120:240:2) -- (C2) arc (240:120:2) -- cycle;
  \end{scope}
  \draw [-latex]   (0,0,0) -- (3,0,0) node [left]  {$x$};
\end{tikzpicture}
\end{document}

