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

Cecilia Wilson

Cecilia Wilson

Answered

2022-11-25

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

Answer & Explanation

artyleriaCuy

artyleriaCuy

Expert

2022-11-26Added 10 answers

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'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 ) ) .

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