Question

Let $\tan \alpha ,\tan \beta$ and $\tan \gamma ;\alpha ,\beta ,\gamma \ne \frac{(2n-1)\pi }{2}$, $n\in N$ 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

Answer: 144

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