Janet Forbes

2022-07-10

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
$F\left(x,y,z\right)={x}^{2}+{y}^{2}-{z}^{2}=1$
at
$\left({x}_{0},{y}_{0},{z}_{0}\right)$
The equation of the tangent plane is
${x}_{0}x+{y}_{0}y-{z}_{0}z=1$

Dalton Lester

Expert

Step 1
The 2 families of skew lines ${L}_{a}$ and ${L}_{b}^{\prime }$ generating hyperboloid with one sheet (H) can be retrieved, starting from its equation

in the following natural way:
Lines
Lines
for any non-zero real number a or b.
Indeed: by multiplication of its 2 equations, (2) $\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}$ (1) ; implication of equations meaning inclusion of corresponding geometric entities ( $\mathrm{\forall }a,{L}_{a}\subset H$ ) as desired. For the same reason, $\mathrm{\forall }b,{L}_{b}^{\prime }\subset H$ .
Therefore, for a given point $\left({x}_{0},{y}_{0},{z}_{0}\right)$ , 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:
$\begin{array}{}\text{(4)}& a=\frac{{y}_{0}-{z}_{0}}{1-{x}_{0}}=\frac{{y}_{0}±\sqrt{{x}_{0}^{2}+{y}_{0}^{2}-1}}{1-{x}_{0}}\end{array}$
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).

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