SOLUTION: Hello, I have a question regarding formulas and equations. So, i tried to figure out this problem, however, I am not sure if the answer is correct: is the graph of 11x - x^2 - 4

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Hello, I have a question regarding formulas and equations. So, i tried to figure out this problem, however, I am not sure if the answer is correct: is the graph of 11x - x^2 - 4      Log On


   

Question 288363: Hello, I have a question regarding formulas and equations.
So, i tried to figure out this problem, however, I am not sure if the answer is correct:
is the graph of 11x - x^2 - 4y^2 = 2y - 16 the graph of an ellipse?
I am not sure...thank you for your time:)
Found 2 solutions by jim_thompson5910, Alan3354:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
11x+-+x%5E2+-+4y%5E2+=+2y+-+16 Start with the given equation.


-x%5E2%2B11x-4y%5E2-2y=-16 Subtract 2y from both sides and rearrange the terms.


-%28x-11%2F2%29%5E2%2B121%2F4-4y%5E2-2y=-16 Complete the square for the x terms. Let me know if you need me to show you how to complete the square.


-%28x-11%2F2%29%5E2%2B121%2F4-4%28y%2B1%2F4%29%5E2%2B1%2F4=-16 Complete the square for the y terms


-%28x-11%2F2%29%5E2-4%28y%2B1%2F4%29%5E2%2B122%2F4=-16 Combine like terms


-%28x-11%2F2%29%5E2-4%28y%2B1%2F4%29%5E2=-16-122%2F4 Subtract 122%2F4 from both sides


-%28x-11%2F2%29%5E2-4%28y%2B1%2F4%29%5E2=-186%2F4 Combine like terms


-%28x-11%2F2%29%5E2-4%28y%2B1%2F4%29%5E2=-93%2F2 Reduce.


%28x-11%2F2%29%5E2%2B4%28y%2B1%2F4%29%5E2=93%2F2 Multiply every term (outside the parenthesis) by -1.


%282%28x-11%2F2%29%5E2%29%2F93%2B%288%28y%2B1%2F4%29%5E2%29%2F93=1 Multiply every term (outside the parenthesis) by 2%2F93 to make the right side equal to 1.


%28%28x-11%2F2%29%5E2%29%2F%2893%2F2%29%2B%28%28y%2B1%2F4%29%5E2%29%2F%2893%2F8%29=1 Rewrite 2%2F93 as 1%2F%2893%2F2%29. Rewrite 8%2F93 as 1%2F%2893%2F8%29.


Now the equation is in the form %28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 where h=11%2F2, k=-1%2F4, a%5E2=93%2F2 and b%5E2=93%2F8 which is an equation of an ellipse.


So %28%28x-11%2F2%29%5E2%29%2F%2893%2F2%29%2B%28%28y%2B1%2F4%29%5E2%29%2F%2893%2F8%29=1 is an ellipse.


Consequently, 11x-x%5E2-4y%5E2=2y-16 is also an ellipse (since the two equations are equivalent).

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
There's an x^2 term and a y^2 term, and their signs are the same.
--> ellipse