Question

ABC is a triangle where $A(2,3,5),B(-1,3,2)$ and $C(\lambda ,5,\mu )$. Let the median through $A$ is equally inclined to the axes.
The value of $\mu -\lambda$ is equal to:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

Now, the mid point of the side $BC$ be $D(\frac{\lambda -1}{2},4,\frac{\mu +2}{2})$.

Let $\alpha$, $\beta$ and $\gamma$ be the angles made by the median $AD$ of $\Delta$ABC.

Then direction cosines of the side $AD$ are

$\cos \alpha =\frac{\frac{\lambda -1}{2}-2}{\overrightarrow{AD}}$, $\cos \beta =\frac{4-3}{\overrightarrow{AD}}$, $\cos \gamma =\frac{\frac{\mu +2}{2}-5}{\overrightarrow{AD}}$.

Since $\alpha =\beta =\gamma$

$\Rightarrow \cos \alpha =\cos \beta =\cos \gamma$......$\because$ (equally inclined to the axes)

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

$\Rightarrow \lambda =7$ and $\mu =10$.

$\therefore \mu -\lambda =3$