SOLUTION: what type of conic is this equation x^2-2y^2+2x+8y-7=0? how can we get the center, the a and the b?

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Question 1120388: what type of conic is this equation x^2-2y^2+2x+8y-7=0? how can we get the center, the a and the b?
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


There is an term and a term. Eliminate Parabola.

The coefficients on and are different. Eliminate a circle.

The signs on the and terms are opposite. Eliminate an ellipse.

Therefore, it must be a hyperbola.

Complete the square:











No conic exists. The graph is two straight lines.


John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
. 
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):


%28x%2B1+-+sqrt%282%29%2A%28y-2%29%29+%2A+%28x%2B1+%2B+sqrt%282%29%2A%28y-2%29%29 = 0,     or, equivalently


 = 0.    (*)


Thus it is the product of the two linear functions

    f(x,y) = x+-+sqrt%282%29%2Ay+%2B+%281%2B2%2Asqrt%282%29%29 

and

    g(x,y) = x+%2B+sqrt%282%29%2Ay+%2B+%281-2%2Asqrt%282%29%29.


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

    x+-+sqrt%282%29%2Ay+%2B+%281%2B2%2Asqrt%282%29%29 = 0      (1)

and/or

    x+%2B+sqrt%282%29%2Ay+%2B+%281-2%2Asqrt%282%29%29 = 0.     (2)


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






Plot  x+-+sqrt%282%29%2Ay+%2B+%281%2B2%2Asqrt%282%29%29 = 0 (red) and plot  x+%2B+sqrt%282%29%2Ay+%2B+%281-2%2Asqrt%282%29%29 = 0 (green)


It is a quite rare case when a hyperbola  DEGENERATES  into the union of its two straight line asymptotes.