The shortest distance between the point [( 3 2,0 )] and the curve y = [ x ], (x 0), is -

Question

The shortest distance between the point  $$\left( {{3 \over 2},0} \right)$$   and the curve y = $$\sqrt x $$,  (x > 0), is -

Accepted Answer

Let points $$\left( {{3 \over 2},0} \right),\left( {{t^2},t} \right),t > 0$$

Distance = $$\sqrt {{t^2} + {{\left( {{t^2} - {3 \over 2}} \right)}^2}} $$

=  $$\sqrt {{t^4} - 2{t^2} + {9 \over 4}}  = \sqrt {{{\left( {{t^2} - 1} \right)}^2} + {5 \over 4}} $$

So minimum distance is $$\sqrt {{5 \over 4}}  = {{\sqrt 5 } \over 2}$$

Solutions · 1