Question

$ABC$ is triangle in which $A=(2,3,5),B=(-1,3,2)$ and $C=(\lambda ,5,\mu )$ . If the median through $A$ is equally inclined to the axes then :

• A. $\lambda =\mu =5$
• B. $\lambda =5,\mu =7$
• C. $\lambda =7,\mu =10$
• D. $\lambda =0,\mu =0$

Answer 1

The correct option is C $\lambda =7,\mu =10$

$D=(\frac{-1+\lambda }{2},\frac{3+5}{2},\frac{2+\mu }{2})$

$D=(\frac{-1+\lambda }{2},4,\frac{2+\mu }{2})$

Direction ratios of $AD$ $=\{\frac{-1+\lambda }{2}-2,4-3,\frac{2+\mu }{2}-5\}$

$=\{\frac{\lambda -5}{2},1,\frac{\mu -8}{2}\}$

$\overrightarrow{AD}=\frac{\lambda -5}{2}\hat{i}+\hat{j}+\frac{\mu -8}{2}\hat{k}$

Now,

$\frac{\lambda -5}{2}=1$

$\Rightarrow$ $\lambda -5=2$

$\therefore$ $\lambda =7$

$\frac{\mu -8}{2}=1$

$\Rightarrow$ $\mu -8=2$

$\Rightarrow$ $\mu =10$

$\therefore$ $\lambda =7$ and $\mu =10$