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.

Answer 1

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

Answer 2

For x = 27 / 4

y = 17 x / 27 = 17 / 4

x + y = 27 / 4 + 17 / 4 = 44 / 4 = 11