math
posted by ayan .
The point (x,y) lies on both conics x2+xy+x=81 and y2+xy+y=51. Given that x+y is positive, determine the value of x+y.

x ^ 2 + x y + x = 81
x ( x + y + 1 ) = 81 Divide both sides by x
x + y + 1 = 81 / x
y ^ 2 + x y + y = 51
y ( y + x + 1 ) = 51
y ( x + y + 1 ) = 51 Divide both sides by y
x + y + 1 = 51 / y
x + y + 1 = x + y + 1
81 / x = 51 / y Multiply both sides by x y
81 x y / x = 51 x y / y
81 y = 51 x Divide both sides by 81
81 y / 81 = 51 x / 81
y = 51 x / 81
y = 3 * 17 * x / ( 3 * 27 )
y = 17 x / 27
Now put this value in formula :
x ( x + y + 1 ) = 81
x ( x + 17 x / 27 + 1 ) = 81
x ( 27 x / 27 + 17 x / 27 + 1 ) = 81
x ( 44 x / 27 + 1 ) = 81
44 x ^ 2 / 27 + x = 81
44 x ^ 2 / 27 + x  81 = 0 Multiply both sides by 27
44 x ^ 2 + 27 x  2187 = 0
The exact solutions of this equation are :
x =  81 / 11 and x = 27 / 4
For x =  81 / 11
y = 17 x / 27 =  51 / 11
x + y =  81 / 11  51 / 11 =  132 / 11 =  12
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4  17 / 4 = 44 / 4 = 11
x + y is positive so :
x = 27 / 4 , y = 17 / 4
x + y = 11 
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4 + 17 / 4 = 44 / 4 = 11
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