SOLUTION: Identify the conic (circle, ellipse, or hyperbola) and state the center 2x^2-y^2+2x+10y-41=0

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Question 81934: Identify the conic (circle, ellipse, or hyperbola) and state the center
2x^2-y^2+2x+10y-41=0
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Start with the given conic

Add 41 to both sides

Group like terms

Factor a 2 out of the first parenthesis
and factor a -1 out of the second parenthesis.

Now we must complete the individual squares inside the parenthesis:

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For instance to complete , take half of 1 and square it (ie to get 0.25. Now add 0.25 inside the parenthesis like this:

Since you really added to the entire left side, we must add to the right side also. Now lets complete

In order to complete , take half of -10 and square it (ie to get 25. Now add 25 inside the parenthesis like this:

Since you really added to the entire left side, we must add to the right side also.

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Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).

Combine like terms on the right side

Factor the individual groups on the left side

Divide both sides by 16.5

Break up the fraction into standard form
So the equation is a hyperbola where the center is (-0.5,5)

From the graph, you can see the center (blue dot)

Here's the graph of
note: the straight lines are not part of the graph. They are the asymptotes with slopes of �.

Here is a check/verification graph of