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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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