What is the least distance between (0,12) and (a(t^2),2at) where a is constant and t is variable.

Let the distance between the given points be , then we know that

 

 

 

 

Let, as when f(t) will be minimum and then D will also be minimum,

 

 

 

,

 

hence f’(t) is monotonically increasing. Minimum of f(t) is when f`(t) reaches zero. The answer thus depends on the value of and cannot be resolved easily in terms of a. So let us say that at t<sub>1</sub> the value of f`(t) is zero and f(t) is minimum, right? Then,

 

Hence, the expression doesn’t resolve to a constant value and the minimum distance will not have a uniform expression for different values of a.