How do I calculate the eccentricity of the conic section below

Question

 Here is the equation 

 

9xy=4

Answer 1

The comment by Mark M is the right path.   

 

The eccentricity is a descriptor of shape.  Since a rotation (without any stretching) does not change shape, a rotation will leave the eccentricity unchanged.   

 

A rotation by 45 degrees is equivalent to the transformation given by:

 

  x = .707 x' - .707 y'

  y = =.707 x'  + .707 y'

 

Where I have written .707 in place of the exact value which is cos(45) = (1/2) sqrt(2)

 

It is easy to work out that x y = (1/2) ( x')²  - (1/2) (y')²   

 

The original equation then becomes   (9/8)  (x')²  - (9/8) (y')²  = 1

 

This is the equation of an East/West hyperbola with a = b = sqrt(8/9)  

 

The eccentricity for a hyperbola in this standard form is sqrt(a² + b²) / a

 

This works out to be sqrt(2)