I want to find the general equation of the two lines of intersection of a...

Janet Forbes

Janet Forbes

Answered

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 ( 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 0 x + y 0 y z 0 z = 1

Answer & Explanation

Dalton Lester

Dalton Lester

Expert

2022-07-11Added 12 answers

Step 1
The 2 families of skew lines L a and L b generating hyperboloid with one sheet (H) can be retrieved, starting from its equation
(1) x 2 + y 2 z 2 = 1             ( y z ) ( y + z ) = ( 1 x ) ( 1 + x ) ,
in the following natural way:
Lines (2) Lines   L a :   { y z = a ( 1 x ) y + z = 1 a ( 1 + x )
Lines (3) Lines   L b :   { y z = b ( 1 + x ) y + z = 1 b ( 1 x )
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 ( a , L a H ) as desired. For the same reason, b , L b 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:
(4) a = y 0 z 0 1 x 0 = y 0 ± x 0 2 + y 0 2 1 1 x 0
which is valid under the condition that x 0 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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