SOLUTION: ihave to write this problem in its standard equation? 36x^2-288x+25y^2-150y=99

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Question 170137: ihave to write this problem in its standard equation?
36x^2-288x+25y^2-150y=99
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
36x%5E2-288x%2B25y%5E2-150y=99 Start with the given equation


36%28x%5E2-8x%29%2B25y%5E2-150y=99 Factor out 36 from the first two terms.


36%28x%5E2-8x%29%2B25%28y%5E2-6y%29=99 Factor out 25 from the last two terms.


Take half of the "x" coefficient -8 to get -8%2F2=-4. Now square -4 to get %28-4%29%5E2=%28-4%29%28-4%29=16


36%28x%5E2-8x%2B16-16%29%2B25%28y%5E2-6y%29=99 Add AND subtract the previous result 16 inside the first parenthesis



Take half of the "y" coefficient -6 to get -6%2F2=-3. Now square -3 to get %28-3%29%5E2=%28-3%29%28-3%29=9


36%28x%5E2-8x%2B16-16%29%2B25%28y%5E2-6y%2B9-9%29=99 Add AND subtract the previous result 9 inside the first parenthesis


36%28%28x%5E2-8x%2B16%29-16%29%2B25%28%28y%5E2-6y%2B9%29-9%29=99 Group the first three terms in both parenthesis


36%28%28x-4%29%5E2-16%29%2B25%28%28y-3%29%5E2-9%29=99 Factor x%5E2-8x%2B16 to get %28x-4%29%5E2. Factor y%5E2-6y%2B9 to get %28y-3%29%5E2


36%28x-4%29%5E2-36%2816%29%2B25%28y-3%29%5E2-25%289%29=99 Distribute


36%28x-4%29%5E2-576%2B25%28y-3%29%5E2-225=99 Multiply


36%28x-4%29%5E2%2B25%28y-3%29%5E2-801=99 Combine like terms.


36%28x-4%29%5E2%2B25%28y-3%29%5E2=99%2B801 Add 801 to both sides.


36%28x-4%29%5E2%2B25%28y-3%29%5E2=900 Combine like terms.


%2836%28x-4%29%5E2%2B25%28y-3%29%5E2%29%2F900=1 Divide both sides by 900. We want the right side to be equal to 1.


%2836%28x-4%29%5E2%29%2F900%2B%2825%28y-3%29%5E2%29%2F900=1 Break up the fraction.


%28%28x-4%29%5E2%29%2F25%2B%28%28y-3%29%5E2%29%2F36=1 Reduce


%28%28x-4%29%5E2%29%2F%285%5E2%29%2B%28%28y-3%29%5E2%29%2F%286%5E2%29=1 Rewrite 25 as 5%5E2. Rewrite 36 as 6%5E2




So now the equation is in standard form %28%28x-h%29%5E2%29%2F%28a%5E2%29%2B%28%28y-k%29%5E2%29%2F%28b%5E2%29=1 (which is an ellipse), where h=4, k=3, a=5, and b=6

This means that the center of the ellipse is (4,3), the length of the minor axis is 2*5=10 units and the length of the major axis is 2*6=12 units


Note: the minor axis is horizontal and the major axis is vertical which means that the ellipse is slightly taller than it is wide.