question_answer

A student moves \[\sqrt{2\,}x\,\text{km}\] East from his residence and then moves x km North. He then goes x km North-East and finally he takes a turn of \[90{}^\circ \] towards right and moves a distance .ic km and reaches his school. What is the shortest distance of the school from his residence?

A) \[3x\,km\]         

B) \[3\sqrt{2}x\,\text{km}\]

C) \[2(\sqrt{2}+1)x\,\text{km}\]      

D) \[2\sqrt{2}\,x\,\text{km}\]

Correct Answer: A

Solution :

In\[\Delta \,BCD,\] \[B{{D}^{2}}=B{{C}^{2}}+C{{D}^{2}}={{x}^{2}}+{{x}^{2}}\] \[\Rightarrow \]   \[BD=\sqrt{2}\,x\] \[\Rightarrow \]   \[BD=AE=\sqrt{2}\,x\] \[\therefore \]    \[OE=OA+AE\]             \[=\sqrt{2}\cdot x+\sqrt{2}\cdot x\]             \[=2\sqrt{2}\,x\] \[\because \]       \[BA=DE=x\] \[\therefore \] In \[\Delta \,ODE,\] \[O{{D}^{2}}=O{{E}^{2}}+D{{E}^{2}}\] \[\therefore \] Minimum distance             \[OD=\sqrt{{{(2\sqrt{2}\cdot x)}^{2}}+{{x}^{2}}}\]             \[=\sqrt{8{{x}^{2}}+{{x}^{2}}}=3x\ \text{km}\]