Find the shortest distance between two parabolas

[cooldudeachyut]

Question : Find the shortest distance between two parabolas y² = x-1 and x² = y-1.

My Attempt : I know that the shortest distance between two curves is along the common normal of the two equations.

I took a point (p,q) on parabola x² = y-1 and (h,k) on parabola y² = x-1

So the equation of normal for x² = y-1 is -> y= -x/2p + 1/2 + q
Equation of normal for y² = x-1 is -> y= -2kx + k(2h+1)

So I get 4 equations :

p² = q-1 ---------1
k² = h-1 ---------2
kp = 1/4 ---------3
2k(2h+1) = (2q+1) ----------4


Using these I wrote shortest distance x = √[(h-p)2 + (k-q)2]

x = √[(k² + 1 + 1/4k)² + (k - 1/16k² -1)²]

I used maxima/minima concept and solved for dx/dk =0

I got the equation = 2(1 + 1/8k³)(2k³ + 3k -3/2 -1/16k²) = 0

Instead of solving the second factor involving quintic equation(Which I do not know how to solve), I solved for the first factor instead.

Solving 1 + 1/8k³ = 0,
I get k = -1/2
h = 5/4
p = -1/2
q = 5/4

So, I get the minimum distance as 7√2/4 but the answer in my textbook is 3√2/4.

So, help me.

 

[ksdhart]

This exact same question was asked at StackExchange. The answerers there have helpfully provided what, at a cursory glance, looks like several different methods of getting to the answer. All of them, however, boil down to the fact that you made a sign error somewhere in your work, as they end up with the closest points on the two parabolas being (1/2, 5/4) and (5/4, 1/2). You can see for yourself, if you wish:

http://math.stackexchange.com/questions/1209893/minimum-distance-between-two-parabolas