Intersection of Hyperboloid & 2-Plane=Ellipse

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I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following:

I'm going to work with $AdS_3$ for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) $X_0^2-X_1^2-X_2^2+X_3^2=L^2$

If I take the eqn of the 2-plane to be (see Figure 11) $X_0+X_2=Le^{w/L}$ then $X_0^2+X_2^2=L^2e^{2w/L}-2X_0X_2$

Substituting for the intersection gives $(X_0+X_2)^2-X_1^2-2X_2^2+X_3^2=L^2 \quad \Rightarrow L^2 e^{2w/L} -2X_0X_2-X_1^2-2X_2^2+X_3^2=L^2$ which I don't recognise as anything to do with an ellipse?

EDIT: solved :)

 

[JorisL]

Please show us what you did.
Also post the figures you reference.

This is common courtesy towards people that find this through google or forum search.