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

Dear Amrit pal

both curve are mirror image to each other about y=x

shortaset distance between these two corve is the segment made by line AB which is perpendicular to both the curve at point A and B respectively

as well as line AB is perpendicular to line y=x ,

i.e  slope of line AB is -1

let a general point A is(t₂ ,t₂² +1)  and point B is (t₁² +1 ,t₁)

 slope of curve x² =y-1 at A is  dy/dx = 2x

                 so slope for perpendicular at point A is -1/2x =-1/2t₂

it will be equal to -1

 -1/2t2 =-1

 t2 =1/2

somilerly slope of perpendiculat on y² =x-1 at point B is  = -2t1

 -2t1 = -1

 t1=1/2

so poins A and B become ( 1/2 ,5/4) and B is (5/4,1/2)

now find the distance between these two point. 

So the minimum distance is (3/4) *√ 2

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