$\text{Assuming } Y \text{ at } (0,0) \text{ to make calculations cleaner :}$
$\text{Using simple trigonometry to find coordinates of } Q \text{ and } X :$
$\text{We have coordinates of two points we can use distance formula :}$
$\text{Q}$ : ($\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$)
$\text{X}$ : ($-\frac{1}{\sqrt{2}}$ , $-1 + \frac{1}{\sqrt{2}}$)
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$\text{So, distance between } Q \text{ and } X ( say \text{ } QX ) \text{ be}:$
$QX = \sqrt{(-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}})^2 + ((-1 + \frac{1}{\sqrt{2}}) - \frac{1}{\sqrt{2}})^2}$
$ = \sqrt{(-\frac{2}{\sqrt{2}})^2 + (-1)^2}$
$ = \sqrt{(-\sqrt{2})^2 + (-1)^2}$
$ = \sqrt{2 + 1}$
$ = \sqrt{3}$
$\boxed{\text{Distance between Q and X is } \sqrt{3} \text{ km}}$
