Question

$A(2,3,5),B(\alpha ,3,3)$ and $C(7,5,\beta )$ are the vertices of a triangle. If the median through $A$ is equally inclined with the co-ordinate axes, then ${\cos }^{-1}\left(\frac{\alpha }{\beta }\right)=$

• A. ${\cos }^{-1}\left(\frac{-1}{9}\right)$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. ${\cos }^{-1}\left(\frac{2}{5}\right)$

Answer 1

Answer: (A)

Given, points $A(2,3,5),B(\alpha ,3,3)$ and $C(7,5,\beta )$
$\therefore$ Mid-point of $BC$ is
$D\left(\frac{\alpha +7}{2},4,\frac{3+\beta }{2}\right)$
$\because$ Direction ratios of line joining points
$A(2,3,5)$ and $D\left(\frac{\alpha +7}{2},4,\frac{3+\beta }{2}\right)$ is
$\left(\frac{\alpha +3}{2},1,\frac{\beta -7}{2}\right)$
$\because$ The line segment $AD$ is equally inclined with the co-ordinate axes, so
$\frac{\alpha +3}{2}=1=\frac{\beta -7}{2}$
$\Rightarrow \alpha =-1\text{ and }\beta =9$
$\therefore {\cos }^{-1}\left(\frac{\alpha }{\beta }\right)$
$={\cos }^{-1}\left(-\frac{1}{9}\right)$