Equation of lines of intersection of tangent plane to a one sheet hyperboloid

I want to find the general equation of the two lines of intersection of a one sheet hyperboloid to its tangent plane for the function

$F(x,y,z)=x^2+y^2-z^2=1$

at

$(x_0,y_0,z_0)$,

The equation of the tangent plane is

$x_0x+y_0y-z_0z=1$

and I've tried substituting this into the hyperboloid equation and I haven't been able to do anything the following mess.

$(z_0^2-x_0^2)x^2+(z_0^2-y_0^2)y^2+2x_0x+2y_0y-2x_0y_0xy-z_0^2-1=0$

I have been provided answers but I don't know how to get there and it's been bugging me for about three days now. How do I get the equation of line of intersection from here?

Solution 1:

Here is a simple approach.

I begin by adapting the information I gave in the following answer some time ago.

The 2 families of skew lines $L_a$ and $L'_b$ generating hyperboloid with one sheet $(H)$ can be retrieved, starting from its equation

$$x^2+y^2-z^2=1 \ \ \ \iff \ \ \ (y-z)(y+z)=(1-x)(1+x),\tag{1}$$

in the following natural way:

$$\text{Lines} \ L_a : \ \begin{cases}y-z&=&a(1-x)\\y+z&=&\dfrac{1}{a}(1+x)\end{cases}\tag{2}$$

$$\text{Lines} \ L'_b : \ \begin{cases}y-z&=&b(1+x)\\y+z&=&\dfrac{1}{b}(1-x)\end{cases}\tag{3}$$

for any non-zero real number $a$ or $b$.

Indeed: by multiplication of its 2 equations, (2) $\implies$ (1) ; implication of equations meaning inclusion of corresponding geometric entities ($\forall a, L_a \subset H$) as desired. For the same reason, $\forall b, L'_b \subset H$.

Therefore, for a given point $(x_0,y_0,z_0)$, you just have to find the values of coefficients $a$ and $b$, which is straightforward.

Consider the case of $a$. From the first equation in (2), one gets:

$$a=\frac{y_0-z_0}{1-x_0}=\frac{y_0 \pm \sqrt{x_0^2+y_0^2-1}}{1-x_0}\tag{4}$$

which is valid under the condition that $x_0 \ne 1$. If $x_0=1$, get $a$ instead from the second equation in (2).

Do the same for $b$ and plug these expressions into (2), resp. (3).