x^2 - 2y^2 + 2x + 8y - 7 = 0.


    Complete the squares:


(x^2 + 2x) - (2y^2 - 8y) = 7             

(x^2 + 2x + 1) - 2*(y^2 - 4y + 4) = 7 + 1 - 8

(x+1)^2 - 2*(y-2)^2 = 0.     (*)



    IT IS NOT a  PARABOLA;  it is not a CIRCLE;  it is not an ELLIPSE;   finally, it is not a HYPERBOLA.


    Then, what is it ?


    It is an equation of two straight lines.



How to see (how to prove) it ?  -- For it, continue with the equation (*)


You see the difference of the two squares in its left side, isn't it ?


Hence, you can rewrite it in this EQUIVALENT form as the product of the sum and the difference of its terms 
    (after taking square roots from them):


 = 0,     or, equivalently


 = 0.    (*)


Thus it is the product of the two linear functions

    f(x,y) =  

and

    g(x,y) = .


Therefore, equation  (*)  represents the UNION  of points, belonging to one of the two straight lines

     = 0      (1)

and/or

     = 0.     (2)


These lines are shown in the figure below: line (1) in red and line (2) in green.






Plot   = 0 (red) and plot   = 0 (green)