find the equation of the hyperbola:
center at (0,0) transverse axis along the x-axis, passing through the points (3,5), (2,-3).
the general equation of such a hyperbola is:
x²/a² - y²/b² = 1 with a and b as the semi-axes.
You know the coordinates of 2 points which must satisfy this equation:
i) 9/a² - 25/b² = 1
ii) 4/a² - 9/b² = 1
From i) you get
iii) a² = (9b²)/(b² + 25).
Plug this term for a² into the equation at ii):
4/((9b²)/(b² + 25)) - 9/b² = 1. Solve for b².
I've got b² = 19/5. Substitute this value into the equation at iii). You'll get
a² = 19/16.
The equation of the hyperbola becomes:
x²/(19/16) - y²/(19/5) = 1 <===> (16x²)/19 - (5y²)/19 = 1