Minimum Value of $x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2$ 
Minimum Value of$\quad$ $x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2$ subject to condition $x_1,y_1$ and $x_2,y_2$ lies on curve $xy=1$. It is also given that  $x_1\gt0$ and $x_2\lt0$

My Approach: $AM\geq GM$
$\frac{x_1^2+y_1^2+x_2^2+y_2^2-2x_1x_2-2y_1y_2}{6}\geq(4x_1^3.y_1^3.x_2^3.y_2^3)^\frac{1}{6}$
and I obtained minimum value as $6\sqrt[3]{2}$.
But I think this is not correct minimum value as when minimum value will occur $AM=GM$ must satisfy and all number must be equal that is $x_1^2=x_2^2=-2x_1x_2=-2y_1y_2=y_1^2=y_2^2$
From first three relation I obtained that $x_1=-2x_2$ and $x_2=-2x_1$  which cannot be true except for zero.
Is my approach correct?

Applying AM-GM, you have to consider if the equality can hold. If not, you only get a strictly lower bound rather than the minimum.
We may use AM-GM in this way:
$$x_1^2 + y_1^2 + x_2^2 + y_2^2 + (- x_1x_2) + (- x_1x_2) + (- y_1 y_2) + (- y_1y_2) 
\ge 8\sqrt[8]{x_1^4 x_2^4 y_1^4 y_2^4}
= 8$$
with equality if $x_1 = y_1 = 1, \ x_2 = y_2 = -1$.
