Given X & # 178; + Y & # 178; + x ^ (- 2) + x ^ (- 2) + y ^ (- 4) - 4 = 0, find (Y & # 178; / X & # 178;) + (X & # 178; / Y & # 178;)

Given X & # 178; + Y & # 178; + x ^ (- 2) + x ^ (- 2) + y ^ (- 4) - 4 = 0, find (Y & # 178; / X & # 178;) + (X & # 178; / Y & # 178;)

If we change the known condition y ^ (- 4) to y (- 2), we can get the known condition as follows: (x-1 / x) 2 + (Y-1 / y) 2 = 0, that is, x = 1 / x, y = 1 / Y; only positive and negative 1 are satisfied, so the formula = 2

X / X-2 - 1 / X & # 178; - 4 = 1! 1111

A:
x/(x-2)-1/(x²-4)=1
Multiply both sides by X & # 178; - 4 to get:
x(x+2)-1=x²-4
x²+2x=x²-4+1
2x=-3
x=-3/2
The test shows that x = - 3 / 2 is the root of the original fractional equation

x/x-2-1=1/x²-4

Multiply both sides by (x + 2) (X-2)
x(x+2) -(x+2)(x-2)=1
x²+2x-x²+4=1
2x=-3
∴x=-3/2
Test: x = - 3 / 2 is the solution of the equation

(x-2/x+2)-(4/x²-4)=1

(x-2)^2/(x^2-4)-4/(x^2-4)=1 [ (x-2)^2-4]=x^2-4 x^2-4x+4=x^2 4x=4 x=1