The shortest distance between the parabolas y^2 =x-1 and x^2=y-1 is?

As shown in the figure due to the symmetry in parabolas the shortest distance AB will be perpendicular to the line x=y
so we just need to find the coordinates of A
tangent at A will be parallel to x=y => have slope 1
For finding that we differentiate P1, and equate the gradient as 1

Thus coordinate of A = (0.5,1.25)
being reflexive about x=y we can say that coordinate of B = (1.25,0.5)
Hence the shortest distance