How do i know if a conic section is a circle, ellipse, hyperbola, or parabola? based on it's equation such as 4x^2 - y^2 + 24x + 4y + 28 = 0

Answer 1

General equation:
ax^2 + by^2 + cx + dy + e = 0

1) if a = b ---> circle
2) if both a and b are positive, but different, ---> ellipse
3) if a is positive and b is negative, ----> hyperbola
4) if either a or b, but not both, are zero ----> parabola
5) if both a and b are zero ---> straight line

so yours is a hyperbola

Answer 2

Find the transformed equation of the hyperbola xy = 4 when rotated 45?

Answer 3

Thanks reiny. This helped a lot.