SOLUTION: 18. Classify the conic section of the equation x^2 + 2x + y^2 + 6y = 15. A circle B ellipse C hyperbola D parabola E point F line I'm not sure how I go about classify

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 18. Classify the conic section of the equation x^2 + 2x + y^2 + 6y = 15. A circle B ellipse C hyperbola D parabola E point F line I'm not sure how I go about classify      Log On


   

Question 162030: 18. Classify the conic section of the equation x^2 + 2x + y^2 + 6y = 15.
A circle
B ellipse
C hyperbola
D parabola
E point
F line

I'm not sure how I go about classifying this. I appericiate any help you have to offer. Thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+%2B+2x+%2B+y%5E2+%2B+6y+=+15 Start with the given equation


x%5E2+%2B+2x+%2Bhighlight%281%29%2By%5E2+%2B+6y+=+15%2Bhighlight%281%29 Take half of the "x" coefficient to get %281%2F2%29%282%29=1. Now square 1 to get 1. Add this value to both sides (on the left, add it right after the "x" term)



x%5E2+%2B+2x+%2B1%2B+y%5E2%2B6y%2Bhighlight%289%29=+15%2B1%2Bhighlight%289%29 Take half of the "y" coefficient to get %281%2F2%29%286%29=3. Now square 3 to get 9. Add this value to both sides (on the left, add it right after the "y" term)


x%5E2+%2B+2x+%2B1%2B+y%5E2+%2B+6y+%2B9=+25 Combine like terms.


%28x%5E2+%2B+2x+%2B1%29%2B+%28y%5E2+%2B+6y+%2B9%29=+25 Group the terms into two groups (with three terms in each group)


%28x%2B1%29%5E2%2B+%28y%5E2+%2B+6y+%2B9%29=+25 Factor x%5E2+%2B+2x+%2B1 to get %28x%2B1%29%5E2


%28x%2B1%29%5E2%2B+%28y%2B3%29%5E2=+25 Factor y%5E2+%2B+6y+%2B9 to get %28y%2B3%29%5E2


%28x-%28-1%29%29%5E2%2B%28y-%28-3%29%29%5E2=+25 Rewrite x%2B1 as x-%28-1%29. Rewrite y%2B3 as y-%28-3%29


%28x-%28-1%29%29%5E2%2B+%28y-%28-3%29%29%5E2=+5%5E2 Replace 25 with 5%5E2


Notice how the equation now fits the form %28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2 where in this case h=-1, k=-3 and r=5. So this shows us that x%5E2+%2B+2x+%2B+y%5E2+%2B+6y+=+15 is a circle. So the answer is A