( h, k ) be any point on the parabola y = x . Let D be the required distance between ( h, k ) and ( , c ). Then D ) h k c h k c ... ( ) Since ( h, k ) lies on the parabola y = x , we have k = h .
So ( ) gives D ≡ D( k ) = k k c D ′ ( k ) = ( k c k k c Now D ′ ( k ) = gives c k Observe that when c k , then ( k c + < , i.e., D ( ) k ′ . Also when c k > , then D ( ) k ′ > . So, by first derivative test, D ( k ) is minimum at c k Hence, the required shortest distance is given by D c c c c c