Given a point on a hyperbolic paraboloid, prove that there exist exactly 2 lines that pass through that point and lie on the surface of that paraboloid.

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soymmernenx · Accepted Answer

Step 1Let                    x        2                    a        2              −                    y        2                    b        2              =  zbe the equation of the Hyperbolic Paraboloid (HP in short).Consider the following family of lines with non zero parameter c  (      L    c    )         {                                                      x              a                                −                                    y              b                                                =                          c                                                                x              a                                +                                    y              b                                                =                                                    z              c                                                  ⟹                      x        2                    a        2              −                    y        2                    b        2              =  z(the implication is obtained by multiplying the equations)But implication means for corresponding sets, inclusion. In this way, we have proven that  (      L    c    )  ⊂  (  H  P  )  .Same reasoning for the other (distinct) family:  (      L    d    ′    )         {                                                      x              a                                +                                    y              b                                                =                          d                                                                x              a                                −                                    y              b                                                =                                                    z              d

Ashly Harrell · Accepted Answer

BTW, The 2 lines are geodesics, unique euclidean straight lines for            x      2              a      2        −            y      2              b      2        =  2  z  .

Given a point on a hyperbolic paraboloid, prove that there

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