SOLUTION: Find the center of the Conic 3x^2+4y^2+18x-32y-5

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Question 83213: Find the center of the Conic
3x^2+4y^2+18x-32y-5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I assume you mean 3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=0

3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=0 Start with the given conic



3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+=+5+Add 5 to both sides



%28+3+%2A+x+%5E+2+%2B+1+8+%2A+x+%29+%2B+%28+4+%2A+y+%5E+2+-+3+2+%2A+y+%29+=+5+ Group like terms




3+%28+x+%5E+2+%2B+6+%2A+x+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%29+=+-+5+ Factor a 3 out of the first parenthesis
and factor a 4 out of the second parenthesis.




Now we must complete the individual squares inside the parenthesis:



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For instance to complete 3+%28+x+%5E+2+%2B+6+%2A+x+%29, take half of 6 and square it (ie %286%2F2%29%5E2=%283%29%5E2=9 to get 9. Now add 9 inside the parenthesis like this:
3+%28+x+%5E+2+%2B+6+%2A+x+%2B+9+%29
Since you really added 3%2A9 to the entire left side, we must add 3%2A9 to the right side also. Now lets complete 4+%28+y+%5E+2+-+8+%2A+y+%29+

In order to complete 4+%28+y+%5E+2+-+8+%2A+y+%29+, take half of -8 and square it (ie %28-8%2F2%29%5E2=%28-4%29%5E2=16 to get 16. Now add 16 inside the parenthesis like this:
4+%28+y+%5E+2+-+8+%2A+y+%2B+1+6+%29+
Since you really added 4%2A16 to the entire left side, we must add 4%2A16 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).

3+%28+x+%5E+2+%2B+6+%2A+x+%2B+9+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%2B+1+6+%29+=+8+6+ Combine like terms on the right side




3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+=+8+6+ Factor the individual groups on the left side

%28+3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+%29+%2F+8+6+=+8+6+%2F+8+6+ Divide both sides by 86




3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+ Break up the fraction into standard form %28x-h%29%5E2%2Fa%5E2%2B%28y-%24k%29%5E2%2Fb%5E2=1

So the equation is an ellipse where the center is (-3,4)
Here's the graph
graph of 3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+ with the center (-3,4)