Question

$ABC$ is a triangle in a plane with vertices $A(2,3,5),B(-1,3,2)$ and $C(\lambda ,5,\mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $({\lambda }^3+{\mu }^3+5)$ is :

• A. 1130
• B. 1348
• C. 676
• D. 1077

Answer 1

Answer: (B)

$D\equiv \left(\frac{-1+\lambda }{2},4,\frac{2+\mu }{2}\right)$
direction cosine of $AD=\left\{\frac{-1+\lambda }{2}-2,4-3,\frac{2+\mu }{2}-5\right\}$
$\left\{\frac{-1+\lambda }{2}-2,4-3,\frac{2+\mu }{2}-5\right\}$
$\overline{AD}=\frac{\lambda -5}{2}i+j+\frac{\mu -8}{2}\hat{k}$
$\Rightarrow \frac{\left(\frac{\lambda -5}{2}\right)}{\sqrt{{\left(\frac{\lambda -5}{2}\right)}^2+1^2+{\left(\frac{\mu -8}{2}\right)}^2}}=\frac{1}{\sqrt{{\left(\frac{\lambda -5}{2}\right)}^2+1+{\left(\frac{\mu -8}{2}\right)}^2}}=\frac{\left(\frac{\mu -8}{2}\right)}{\sqrt{{\left(\frac{\lambda -5}{2}\right)}^2+1+{\left(\frac{\mu -8}{2}\right)}^2}}$
$\overline{AD}\cdot i=\overline{AD}\cdot \hat{j}=\overline{AD}\cdot \hat{k}$
$\lambda =7,\mu =10$
${\lambda }^3+{\mu }^3+5=343+1000+5=1348$