Question 93595
Given equation: {{{x^2 - 4y^2 - 4x - 24y - 48 = 0}}}


Compare with the standard equation for a conic
{{{ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0}}}
If {{{h^2 - ab = 0}}} then the conic is a parabola.
If {{{h^2 - ab < 0}}} then the conic is an ellipse.
If {{{h^2 - ab > 0}}} then the conic is a hyperbola.
If {{{a = b}}} and {{{h = 0}}} then the conic is a circle.


Here, a = 1, b = -4, h = 0.
So {{{h^2 - ab = 4 > 0}}} and hence the conic is a hyperbola.