Find the minimum distance between the curves y^2 = x-1 and x^2 = y-1
Please tell the procedure, how to solve this?
The points on the parabolas where the tangents have gradient 1 are on , and on . The distance between these points is .
In general, if you can't use that sort of symmetry to simplify the problem, you would have to find the equation of the normal at a general point on one curve. Then find the point(s), if any, where that line meets the other curve, determine the distance along the normal to the closest point on the other curve, and finally minimise that distance.
Edit. For the general case, Mr F's method (see next comment) is much better than my laborious procedure of using normals.
However ..... here's a hint for an easier way: note that the two curves are inverses of each other and are therefore reflections of each other in the line y = x.
Edit: I was very slow. Beaten by Opalg.