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

Both the parabolas are symmetric to the line x=y.

Therefore, equating their slopes to 1,

for parabola x^2=y-1,

dy/dx = 1

2x =1

x=1/2

corresponding value of y by putting the value of x in the eq. of parabola we get y=5/4

Similarly, for the parabola y^=x-1,

dx/dy =1

2y=1

y=1

corresponding value of x by putting the value of y in the eq. of parabola we get x=5/4

The minimum distance is between the points (1/2,5/4) and (5/4,1/2)

And 3√2/4