It looks good to me. Of course, I can't speak for your teacher!
The problem is:
"Parametrize the intersection of the hyperboloid and the plane , where is a positive real number. Give the range of the values taken by the parameter. -Hint: Consider the cases 0<a<1 and a > 1 separately, and remember that x, y and z are all real."
So here's what I did:
I first plugged in the equation of the plane into the equation of the hyperboloid:
If a<1, I get an ellipse of the form: , similar to
From an online source, I found that and , and therefore and that
If a>1, the equation can be put in the form which is the equation of a hyperbola. Once again, from an online source (Wolfram MathWorld), I found that I could represent both branches of the hyperbola with , and , where with discontinuities at
I've seriously spent a lot of time on this problem, and this is the best I can come up with. Can someone tell me if it's acceptable? If not, what should I do?
I don't see why you have to do all that. If I substitute the plane into the hyperboloid, I get for :
so that and . So a parametric representation of the intersection is:
with for
and:
for
And I think for the case , then which then means so that the paramaterization is .
The yellow contours below is this paramaterization for a=1/2 and a=2
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}]