Ok, I think that should be when . Also then for . Also, I drew that in Mathematica using this code for the a=1/2 one:

Code:

a = 1/2;
p1 = ParametricPlot3D[{x, Sqrt[1/(1 - a^2) - x^2/(1 - a^2)],
a*Sqrt[1/(1 - a^2) - x^2/(1 - a^2)]}, {x, -1, 1},
PlotStyle -> {Thickness[0.008], Yellow}]
p2 = ParametricPlot3D[{x, -Sqrt[1/(1 - a^2) - x^2/(1 - a^2)],
(-a)*Sqrt[1/(1 - a^2) - x^2/(1 - a^2)]}, {x, -1, 1},
PlotStyle -> {Thickness[0.008], Yellow}]
pic1 = Show[{ContourPlot3D[{x^2 + y^2 == z^2 + 1, z == (1/2)*y}, {x, -3, 3},
{y, -3, 3}, {z, -3, 3}], p1, p2}]