Shortest distance between the two parabolas y² = x – 2, x² = y – 2 is

$\begin{array}{l}Given parabolas y^2=x-2\to \left(i\right),x^2=y-2\to \left(ii\right) are symmetrical about the line y=x\\ slope of the \tan gent parallel to y=x is 1\\ \Rightarrow 2y\frac{dy}{dx}=1\Rightarrow y=\frac{1}{2}\\ from \left(i\right) x=\frac{9}{4}\\ P(x,y)=(\frac{1}{2},\frac{9}{4})\\ Point on second parabola=(\frac{9}{4},\frac{1}{2})\\ \\ \Rightarrow PQ=\frac{7}{2\sqrt{2}}\end{array}$