Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty. Enter the most specific answer.
y^2 - x^2 - x +5y - 25 = 0
Rearrange as
y^2 + 5y - x^2 - x = 25 complete the square on x and y
y^2 + 5y + 25/4 - ( x^2 + x + 1/4) = 25 + 25/4 - 1/4
(y + 5/2)^2 - ( x + 1/2)^2 = 31 divide through by 31
( y + 5/2)^2 (x + 1/2)^2
_________ - __________ = 1
31 31
This is a hyperbola
