A question about hyperbolic functions Suppose $(x,y,z),(a,b,c)$ satisfy $$x^2+y^2-z^2=-1, z\ge 1,$$ $$ax+by-cz=0,$$ $$a^2+b^2-c^2=1.$$ Does it follow that $$z\cosh(t)+c\sinh(t)\ge 1$$ for all real number $t$?

The curve  $(X_1,X_2,X_3)=\cosh(t)(x,y,z)+\sinh(t)(a,b,c), -\infty<t<\infty$ is continuous and satisfies $X_1^2+X_2^2-X_3^2=-\cosh^2(t)+\sinh^2(t)=-1$. One of its point $(x,y,z)$ (when $t=0$) lies on the upper sheet $X_1^2+X_2^2-X_3^2=-1, X_3\ge 1$. By connectness of the curve, the whole curve must lie in this connected component. Hence $z\cosh(t)+c\sinh(t)\ge 1$ for all $t$.
