Question 176404
{{{25x^2+16y^2+150x=160y-225}}} Start with the given equation.



{{{25x^2+16y^2+150x-160y=-225}}} Subtract 160y from both sides.



{{{(25x^2+150x)+(16y^2-160y)=-225}}} Group like terms.



{{{25(x^2+6x)+(16y^2-160y)=-225}}} Factor 25 from the first group (to make the {{{x^2}}} coefficient equal to 1)



{{{25(x^2+6x)+16(y^2-10y)=-225}}} Factor 16 from the second group (to make the {{{y^2}}} 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(x^2+6x+9-9)+16(y^2-10y)=-225}}} Add AND subtract 9 in the first parenthesis.



{{{25((x+3)^2-9)+16(y^2-10y)=-225}}} Factor {{{x^2+6x+9}}} to get {{{(x+3)^2}}}



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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((x+3)^2-9)+16(y^2-10y+25-25)=-225}}} Add AND subtract 25 in the second parenthesis.



{{{25((x+3)^2-9)+16((y-5)^2-25)=-225}}} Factor {{{y^2-10y+25}}} to get {{{(y-5)^2}}}



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{{{25(x+3)^2-25(9)+16(y-5)^2-16(25)=-225}}} Distribute



{{{25(x+3)^2-225+16(y-5)^2-400=-225}}} Multiply



{{{25(x+3)^2+16(y-5)^2-625=-225}}} Combine like terms.



{{{25(x+3)^2+16(y-5)^2=-225+625}}} Add 625 to both sides.



{{{25(x+3)^2+16(y-5)^2=400}}} Combine like terms.



{{{(25(x+3)^2+16(y-5)^2)/400=cross(400/400)}}} Divide both sides by 400 (to make the right side equal to 1)



{{{(25(x+3)^2)/400+(16(y-5)^2)/400=1}}} Break up the fraction.



{{{((x+3)^2)/16+((y-5)^2)/25=1}}} Reduce



{{{((x+3)^2)/(4^2)+((y-5)^2)/(5^2)=1}}} Rewrite 16 as {{{4^2}}}. Rewrite 25 as {{{5^2}}}



{{{((x-(-3))^2)/(4^2)+((y-5)^2)/(5^2)=1}}} Rewrite {{{x+3}}} as {{{x-(-3)}}}



Now the equation is in the form {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}} (which is the standard form of an ellipse) where {{{h=-3}}}, {{{k=5}}}, {{{a=4}}} and {{{b=5}}}



So the value of {{{a^2}}} is {{{a^2=4^2=16}}}