Find the minimum distance between the curves y^2 = x-1 and x^2 = y-1 please give detail solution

Interesting problem.

I found the distance is from point ( 5/4 , 1/2 ) to point ( 1/2, 5/4 )

Note the symmetry of the curve.

So the minimum distance is (3/4) * sqrt (2 ) which is approximately 1.06

I will come back with more details.

My general approach is this:

let the points be (x1,y1) and (x2,y2 )

Find the distance square

d^2 = (x1 -x2) ^2 + (y1 -y2)^2

eliminate (y1, y2 ) from d^2

set ( d (d^2) / dx1 =0 , d (d^2) /dx2 =0 )

solve for (x1,x2)

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regards

Ran\mesh