I want to find the general equation of the two lines of intersection of a...
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
The equation of the tangent plane is
Answer & Explanation
The 2 families of skew lines and generating hyperboloid with one sheet (H) can be retrieved, starting from its equation
in the following natural way:
for any non-zero real number a or b.
Indeed: by multiplication of its 2 equations, (2) (1) ; implication of equations meaning inclusion of corresponding geometric entities ( ) as desired. For the same reason, .
Therefore, for a given point , 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:
which is valid under the condition that . If , get a instead from the second equation in (2).
Do the same for b and plug these expressions into (2), resp. (3).