Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. x^2
 − 4y^2 − 2x
− 16y − 31 = 0

Answer 1

http://www.wolframalpha.com/input/?i=x%5E2-4y%5E2-2x-16y-31%3D0

So I used Wolfram to plot it out for me, so it's an obvious hyperbola, but the q also asks to justify for full credit, so I guess there is some work involved....

Answer 2

compare it with .... http://en.wikipedia.org/wiki/Conic_section#Discriminant_classification
you will know what it is.

but by observation i can say that it's a hyperbola.
because i can complete square , (x- h)^2 - (y-k)^2 = 0 <-- probably this is easier method ... but does not work sometimes.