Point P on curve y = cosh x is such that its perpendicular distance from the line y = x is a minimum. Show P's coordinates are (ln(1 + root 2), root 2)

Question

 Really stuck on this, any help appreciated :)

Answer 1

The distance from a point on y = cosh x to a point on y = x is given by

 

d = (1/√2)(x - cosh x)

 

The derivative of d with respect to x = (1/√2)(1 - sinh x)

and the minimum will occur when sinh x = 1.

 

Since sinh^-1x = ln[x + sqrt(x² + 1)], sinh x = 1 when x = ln(1 + √2)

 

and since cosh²x - sinh²x = 1, when sinh x = 1,  cosh x =  √2

 

all of that means that the minimum point is as stated in the problem (ln(1 + √2),√2)