question_answer

The projection of any line on coordinate axes be respectively \[3,\,\,\,4,\,\,\,5,\] then its length is:

A) \[12\]                                   

B) \[50\]

C) \[5\sqrt{2}\]                                      

D)   none of these

Correct Answer: C

Solution :

Key Idea: If\[\cos \alpha ,\,\,\cos \beta \]and\[\cos \gamma \]are the direction cosines of a line, then                 \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\] In\[\Delta \,ABC\],                 \[\cos \alpha =\frac{3}{AB}\] In\[\Delta \,\,ABD\],                 \[\cos \beta =\frac{4}{AB}\] And in\[\Delta \,ABE\],                 \[\cos \gamma =\frac{5}{AB}\] We know,                 \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\] \[\Rightarrow \]               \[{{\left( \frac{3}{AB} \right)}^{2}}+{{\left( \frac{4}{AB} \right)}^{2}}+{{\left( \frac{5}{AB} \right)}^{2}}=1\] \[\Rightarrow \]               \[A{{B}^{2}}=9+16+25\] \[\Rightarrow \]               \[A{{B}^{2}}=50\] \[\Rightarrow \]