Tangent Space of Singel layered hyperboloid

[Mr.Brown]

Tangent space of single layered hyperboloid

Ok i´m given a single layered hyperboloid given by $$\left(\frac{x}{a}-\frac{z}{c}\right)\cdot\left(\frac{x}{a}+\frac{z}{c}\right)-\left(1-\frac{y}{b}\right)\cdot\left(1+\frac{y}{b}\right)=0$$

Now the Problem asks me to take this as a vanishing determinant, i guess they mean

$$\begin{vmatrix} \frac{x}{a}-\frac{z}{c} &1-\frac{y}{b} \\ 1+\frac{y}{b} & \frac{x}{a}+\frac{z}{c} \end{vmatrix}=0$$


Now out of the knowledge that the surface is parametrized by a vanishing determinant i should find 2 streight lines through every point p in the surface.

Now I am totally lost how to do it by the determinant.

Im perfectly fine with just taking the tangent space at p and the just look for the direction vectors that produce lines entirely in the surface but how to do it this way.

I thought of i as a Jacobian of some function (of a,b,c ?? )for whitch the hyperboloid is sort of a level set but i don´t know how to go on sorry some hint would be fine :)
Thanks

 

[Mr.Brown]

come on you guys :)

 

[Mr.Brown]

Ok solved it :=)