question_answer

Let A (2, 3, 5), B (? 1, 3, 2) and \[C(\lambda ,5,\mu )\]be the vertices of a DABC. If the median through A is equally inclined to the coordinate axes, then:   [JEE Main Online Paper ( Held On 11 Apirl  2014 )

A) \[5\lambda -8\mu =0\]                 

B) \[8\lambda -5\mu =0\]

C) \[10\lambda -7\mu =0\]              

D) \[7\lambda -10\mu =0\]

Correct Answer: C

Solution :

If D be the mid-point of BC, then \[D=\left( \frac{\lambda -1}{2},4,\frac{\mu +2}{2} \right)\] Direction ratios of AD are \[\frac{\lambda -5}{2},1,\frac{\mu -8}{2}\] Since median AD is equally inclined with coordinate axes, therefore direction ratios of AD will be equal, i.e,\[\frac{{{\left( \frac{\lambda -5}{2} \right)}^{2}}}{{{\left( \frac{\lambda -5}{2} \right)}^{2}}+1+{{\left( \frac{\lambda -8}{2} \right)}^{2}}}\] \[=\frac{1}{{{\left( \frac{\lambda -5}{2} \right)}^{2}}+1+{{\left( \frac{\mu -8}{2} \right)}^{2}}}\] \[\Rightarrow \]\[{{\left( \frac{\lambda -5}{2} \right)}^{2}}=1={{\left( \frac{\mu -8}{2} \right)}^{2}}\] \[\Rightarrow \]\[\lambda =7,3\]and\[\mu =10,6\] If\[\lambda =7\]and\[\mu =10\] Then\[\frac{\lambda }{\mu }=\frac{7}{10}\Rightarrow 10\lambda -7\mu =0\]