Intersecting Lines on a Hyperbolic Paraboloid

Question

I have a hyperbolic paraboloid with the equation $f(x,y) = x^2-y^2 $

Given a point $P = (-1, \frac{1}{2}, \frac{3}{4}) $ , how do I find the two intersecting straight lines that lie on the surface and go through P?

Answer 1

The two systems of lines $\ell_a$ and $L_b$ making hyperbolic paraboloid $(HP)$ (with equation $z=x^2-y^2)$ a "doubly ruled" surface have the following equations :

$$\ell_a \ \begin{cases}x-y=\dfrac{z}{a}\\x+y=a\end{cases}$$

$$ L_b \ \begin{cases}x-y=b\\x+y= \dfrac{z}{b}\end{cases}$$

It suffices then to express that $P$ belongs to each of the two families by replacing $x,y,z$ by its coordinates to get a (unique) value for $a$ and for $b$.

Remark : the fact that for example $\ell_a$ is a family of lines belonging to the $(HP)$ is due to the implication (by taking the product of equations) :

$$ \begin{cases}x-y=\dfrac{z}{a}\\x+y=a\end{cases} \ \implies x^2-y^2=z$$

and one knows that properties expressed as implications corresponds to set inclusions, here $\ell_a \subset (HP)$.