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 = 1at ( x 0 , y 0 , z 0 )The equation of the tangent plane is x 0 x + y 0 y − z 0 z = 1

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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    =  1at  (      x    0    ,      y    0    ,      z    0    )The equation of the tangent plane is      x    0    x  +      y    0    y  −      z    0    z  =  1

Dalton Lester · Accepted Answer

Step 1The 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).

I want to find the general equation of the two lines of intersection of a one sheet hyperboloid to i

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