Question

$△ABC$ has vertices at $A\equiv (2,3,5),B\equiv (-1,3,2)$ and $C\equiv (\lambda ,5,\mu )$. If the median through $A$ is equally inclined to the axes, then the values of $\lambda$ and $\mu$ respectively are.

• A. $10,7$
• B. $9,10$
• C. $7,9$
• D. $7,10$

Answer 1

The correct option is D $7,10$

$D\equiv (\frac{\lambda -1}{2},\frac{5+3}{2},\frac{\mu +2}{2})$
$\equiv (\frac{\lambda -1}{2},4,\frac{\mu +2}{2})$
Direction ratios of $AD$ are $\frac{\lambda -1}{2}-2,4-3,\frac{\mu +2}{2}-5$
i.e. $\frac{\lambda -1-4}{2},1,\frac{\mu +2-10}{2}$
i.e. $\frac{\lambda -5}{2},1,\frac{\mu -8}{2}$
Since the line $AD$ is equally inclined to coordinate axes, its direction ratios are in ratio $\pm 1:\pm 1:\pm 1$

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

$⟹\lambda -5=2$

$⟹\lambda =7$

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

$⟹\frac{\mu -8}{2}=1$

$⟹\mu =10$

Hence, the values of $\lambda$ and $\mu$ are $7$ and $10$ respectively.