How do I parameterize the intersection of x 2 − y 2 = 11 and z = x y

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How do I parameterize the intersection of       x    2    −      y    2    =  11 and   z  =  x  y

artyleriaCuy · Accepted Answer

Use   sec and   tan, or   cosh and   sinh instead of   cos and   sin. You just need to remember that   1  +      tan    2    ⁡  t  =      sec    2    ⁡  t, or that       cosh    2    ⁡  t  −      sinh    2    ⁡  t  =  1. I&#039;ll go with the hyperbolic functions. Since       x    2    −      y    2    =  11 is equivalent to   (  x      /        11        )    2    −  (  y      /        11        )    2    =  1, we may simply let   x  =      11    cosh  ⁡  t and   y  =      11    sinh  ⁡  t. The second equation immediately gives z. One possible parametrization is  t  ↦  (      11    cosh  ⁡  t  ,      11    sinh  ⁡  t  ,  (  11      /    2  )  sinh  ⁡  (  2  t  )  )  ,for example. Observe that the intersection is not connected, and the other half is parametrized by  t  ↦  (  −      11    cosh  ⁡  t  ,      11    sinh  ⁡  t  ,  −  (  11      /    2  )  sinh  ⁡  (  2  t  )  )  .

How do I parameterize the intersection of x^2−y^2=11 and z=xy

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