Question

Let $\tan \alpha, \tan \beta$ and $\tan\gamma; \alpha, \beta, \gamma \neq \frac{(2 n -1) \pi}{2}$, $n \in N$ be the slopes of three line segments $OA$, $OB$ and $OC$, respectively, where $O$ is origin.If circumcentre of $\Delta ABC$ coincides with origin and its orthocentre lies on y-axis, then the value
of $\left(\frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^{2}$ is equal to :

Answer 1

Since orthocentre and circumcentre both lies on $y$ -axis
$\Rightarrow $ Centroid also lies on $y$ -axis
$\Rightarrow \Sigma \cos \alpha=0 %$
$ \cos \alpha+\cos \beta+\cos \gamma=0 $
$\Rightarrow \cos ^{3} \alpha+\cos ^{3} \beta+\cos ^{3} \gamma=3 \cos \alpha \cos \beta \cos \gamma $
$\therefore \frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}$
$= \frac{4\left(\cos ^{3} \alpha+\cos ^{3} \beta+\cos ^{3} \gamma\right)-3(\cos \alpha+\cos \beta+\cos \gamma)}{\cos \alpha \cos \beta \cos \gamma}$
$=12 $