Question

the transverse axis of a hyperbola lies along the x-axis and its center is at the origin. the length of the conjugate axis is 8 and the point (3,3) lies on the hyperbola.

find its eccentricity and the length of latus rectum.

Answer 1

As the length of the conjugate axis is 8 , so 2b = 8 or b = 4
And the transverse axis is along x -axis, so the equation of hyperbola will be:
x²/a² - y²/b² =1
And (3,3) lies on the hyperbola, so it will satisfy the equation.
Hence 9/a² - 9/16 = 1
Or a² = 144/25
And the eccentricity
b² = a² (e²-1)
So 16 = (144/25)(e² -1)
Or e = root(34)/3

And latus rectum = 2b²/a²
So latus rectum = 2*16/(144/25) = 50/9