question_answer

A helicopter is flying along the curve given by\[y-{{x}^{3/2}}\,=7,\,\,(x\ge 0)\,\].   A soldier positioned at the point \[\left( \frac{1}{2},\,\,7 \right)\]wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is: [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

A) \[\frac{1}{6}\,\sqrt{\frac{7}{3}}\]                                  

B) \[\frac{\sqrt{5}}{6}\]

C) \[\frac{1}{2}\]              

D)                  \[\frac{1}{3}\,\sqrt{\frac{7}{3}}\]

Correct Answer: A

Solution :

\[y-{{x}^{3/2}}=7\left( x\ge 0 \right)\] \[\frac{dy}{dx}\,=\frac{3}{2}{{x}^{1/2}}\] \[\left( \frac{3}{2}\,\sqrt{x} \right)\left( \frac{7-y}{\frac{1}{2}-x} \right)\,=\,-1\] \[\frac{3}{2}{{x}^{2}}\,=\,\,\frac{1}{2}-x\] \[\left( x+1 \right)\left( 3x-1 \right)=0\] \[x=-1\] (rejected) \[x=\frac{1}{3}\] \[y=7+{{x}^{3/2}}=7+{{\left( \frac{1}{3} \right)}^{3/2}}\] \[{{\ell }_{AB}}=\,\sqrt{{{\left( \frac{1}{2}-\frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{3} \right)}^{3}}}\]             \[=\,\,\sqrt{\frac{7}{108}}\,=\,\frac{1}{6}\sqrt{\frac{7}{3}}\]