question_answer

Given, two points \[A\equiv (-2,\,0)\] and\[B\equiv (0,\,\,4)\]. The coordinates of a point M lying on the line \[y=x\], so that the perimeter of the \[\Delta \,AMB\] is least, is

A)  \[(1,\,\,1)\]                       

B)  \[(0,\,\,0)\]

C)  \[(9,\,\,2)\]                       

D)  \[(3,\,\,3)\]

Correct Answer: B

Solution :

 For perimeter to be minimum, AM + BM should be minimum.         \[\frac{1}{{{k}_{s}}}=\frac{1}{k}\,\left( \frac{1}{1-1/2} \right)\] AB is fixed) For AM + BM to be minimum. M should be such that AM is reflected along MB from the line \[\Rightarrow \] \[\frac{1}{{{k}_{s}}}=\frac{2}{k}\Rightarrow \,{{k}_{s}}=\frac{k}{2}\] [C is reflection of A on line \[\Rightarrow \] i.e. \[T=2\pi \sqrt{\frac{m}{{{k}_{s}}}}\,\Rightarrow \,T=2\pi \sqrt{\frac{2m}{k}}\] So, the equation of CB is \[{{f}_{approach}}=\left( \frac{v}{v-{{v}_{s}}} \right)\,{{f}_{0}}={{f}_{0}}={{f}_{1}}\]. \[{{f}_{recede}}=\left( \frac{v}{v+{{v}_{s}}} \right){{f}_{0}}={{f}_{2}}\,(<{{f}_{1}})\]         \[{{Q}_{1}}+(-{{W}_{1}})={{Q}_{2}}+(-{{W}_{2}})={{U}_{B}}-{{U}_{A}}\]