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 $B C$ 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 $A D$ is equally inclined with the co-ordinate axes, so
$\frac{\alpha+3}{2} =1=\frac{\beta-7}{2}$
$\Rightarrow \alpha=-1 \text { and } \beta =9$
$\therefore \cos ^{-1}\left(\frac{\alpha}{\beta}\right)$
$=\cos ^{-1}\left(-\frac{1}{9}\right)$