Question

Let $A(2,3,5),B(-1,3,2),C(\lambda ,5,\mu )$ are the vertices of a triangle and its median through $A,AD$ is equally inclined to the coordinate axes. Projection of $\text{AB on BC is:}$

• A. $\frac{24}{\sqrt{33}}$
• B. $\frac{8\sqrt{3}}{11}$
• C. $-48$
• D. $48$

Answer 1

The correct option is A $\frac{24}{\sqrt{33}}$


$A(2,3,5),B(-1,3,2),C(\lambda ,5,\mu )$
$D=\text{mid point of}\overline{BC}=(\frac{\lambda -1}{2},4,\frac{\mu +2}{2})$
$D{R}^{\prime }s\text{of}\overline{AD}=(\frac{\lambda -5}{2},1,\frac{\mu -8}{2})$
$D.R^{\prime }S\overline{AD}\text{is equally inclined to axes}$
$\frac{\lambda -5}{2}=1=\frac{\mu -8}{2}\Rightarrow \lambda =7\Rightarrow \mu =10\Rightarrow 2\lambda -\mu =4$
$\text{Projection of}\overline{AB}\text{on}\overline{BC}=\frac{\overline{AB}.\overline{BC}}{|\overline{BC}|}$
We have $\overline{AB}=(-3,0,-3),\overline{BC}=(8,2,8)$
Hence Projection of $\overline{AB}\text{on}\overline{BC}=|\frac{-24+0-24}{\sqrt{132}}|=\frac{24}{\sqrt{33}}$