SOLUTION: I need help putting 25x^2+16y^2+150x=160y-225 in standard form and then finding the a^2 value.

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Question 176404: I need help putting 25x^2+16y^2+150x=160y-225 in standard form and then finding the a^2 value.
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
I need help putting 25x^2+16y^2+150x=160y-225 in standard form and then finding the a^2 value.
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Complete the square on the x-terms and on the y-terms:
25(x^2 + 6x + ?) + 16(y^2 - 10y + ?) = -225
25(x^2 + 6x + 9) + 16(y^2 - 10y + 25) = -225 + 25*9 + 16*25
25(x+3)^2 + 16(y-5)^2 = 400
[(x+3)^2]/16 + [(y-5)^2]/25 = 1
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a^2 = 25
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Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
25x%5E2%2B16y%5E2%2B150x=160y-225 Start with the given equation.


25x%5E2%2B16y%5E2%2B150x-160y=-225 Subtract 160y from both sides.


%2825x%5E2%2B150x%29%2B%2816y%5E2-160y%29=-225 Group like terms.


25%28x%5E2%2B6x%29%2B%2816y%5E2-160y%29=-225 Factor 25 from the first group (to make the x%5E2 coefficient equal to 1)


25%28x%5E2%2B6x%29%2B16%28y%5E2-10y%29=-225 Factor 16 from the second group (to make the y%5E2 coefficient equal to 1)


Take half of the "x" coefficient 6 to get 3. Square 3 to get 9. Add AND subtract this value inside the first parenthesis:


25%28x%5E2%2B6x%2B9-9%29%2B16%28y%5E2-10y%29=-225 Add AND subtract 9 in the first parenthesis.


25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%29=-225 Factor x%5E2%2B6x%2B9 to get %28x%2B3%29%5E2


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Take half of the "y" coefficient -10 to get -5. Square -5 to get 25. Add AND subtract this value inside the second parenthesis:


25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%2B25-25%29=-225 Add AND subtract 25 in the second parenthesis.


25%28%28x%2B3%29%5E2-9%29%2B16%28%28y-5%29%5E2-25%29=-225 Factor y%5E2-10y%2B25 to get %28y-5%29%5E2


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25%28x%2B3%29%5E2-25%289%29%2B16%28y-5%29%5E2-16%2825%29=-225 Distribute


25%28x%2B3%29%5E2-225%2B16%28y-5%29%5E2-400=-225 Multiply


25%28x%2B3%29%5E2%2B16%28y-5%29%5E2-625=-225 Combine like terms.


25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=-225%2B625 Add 625 to both sides.


25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=400 Combine like terms.


%2825%28x%2B3%29%5E2%2B16%28y-5%29%5E2%29%2F400=cross%28400%2F400%29 Divide both sides by 400 (to make the right side equal to 1)


%2825%28x%2B3%29%5E2%29%2F400%2B%2816%28y-5%29%5E2%29%2F400=1 Break up the fraction.


%28%28x%2B3%29%5E2%29%2F16%2B%28%28y-5%29%5E2%29%2F25=1 Reduce


%28%28x%2B3%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1 Rewrite 16 as 4%5E2. Rewrite 25 as 5%5E2


%28%28x-%28-3%29%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1 Rewrite x%2B3 as x-%28-3%29


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


So the value of a%5E2 is a%5E2=4%5E2=16