JEE MAIN - Mathematics (2021 - 17th March Evening Shift - No. 16)
Let tan$$\alpha$$, tan$$\beta$$ and tan$$\gamma$$; $$\alpha$$, $$\beta$$, $$\gamma$$ $$\ne$$ $${{(2n - 1)\pi } \over 2}$$, 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 $${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2}$$ is equal to ____________.
Answer
144
Explanation
Since orthocentre and circumcentre both lies on y-axis.
$$ \Rightarrow $$ Centroid also lies on y-axis.
$$ \Rightarrow $$ $$\sum {\cos \alpha = 0} $$
cos$$\alpha$$ + cos$$\beta$$ + cos$$\gamma$$ = 0
$$ \Rightarrow $$ cos3 $$\alpha$$ + cos3 $$\beta$$ + cos3 $$\gamma$$ = 3cos$$\alpha$$cos$$\beta$$cos$$\gamma$$
$$ \therefore $$ $${{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}$$
$$ = {{4({{\cos }^3}\alpha + {{\cos }^3}\beta + {{\cos }^3}\gamma ) - 3(\cos \alpha + \cos \beta + \cos \gamma )} \over {\cos \alpha \cos \beta \cos \gamma }} = 12$$
then, $${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2} = 144$$
$$ \Rightarrow $$ Centroid also lies on y-axis.
$$ \Rightarrow $$ $$\sum {\cos \alpha = 0} $$
cos$$\alpha$$ + cos$$\beta$$ + cos$$\gamma$$ = 0
$$ \Rightarrow $$ cos3 $$\alpha$$ + cos3 $$\beta$$ + cos3 $$\gamma$$ = 3cos$$\alpha$$cos$$\beta$$cos$$\gamma$$
$$ \therefore $$ $${{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}$$
$$ = {{4({{\cos }^3}\alpha + {{\cos }^3}\beta + {{\cos }^3}\gamma ) - 3(\cos \alpha + \cos \beta + \cos \gamma )} \over {\cos \alpha \cos \beta \cos \gamma }} = 12$$
then, $${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2} = 144$$
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