Calculus is not needed II

Geometry Level 4

$\sqrt { { \alpha }^{ 2 }-4 } \times \tan { A } +\alpha \times \tan { B } +\sqrt { { \alpha }^{ 2 }+4 } \times \tan { C } =6\alpha$

If $\alpha$ is real constant and $A, B$ and $C$ are variable angles satisfying the condition above, then find the least value of $\tan ^{ 2 }{ A } +\tan ^{ 2 }{ B } +\tan ^{ 2 }{ C }$

The answer is 12.

1 solution

Tanuj Ravi
Jan 18, 2016

The condition can be written as

$(\sqrt { { \alpha }^{ 2 }-4 } i+\alpha j+\sqrt { { \alpha }^{ 2 }+4 } k).(\tan { A } i+\tan { B } j+\tan { C } k)=6\alpha )$

$\sqrt { ({ \alpha }^{ 2 }-4)+{ \alpha }^{ 2 }+({ \alpha }^{ 2 }+4) } \times \sqrt { \tan ^{ 2 }{ A } +\tan ^{ 2 }{ B } +\tan ^{ 2 }{ C } } \times \cos { \beta } =6\alpha$

Simplifying

$\tan ^{ 2 }{ A } +\tan ^{ 2 }{ B } +\tan ^{ 2 }{ C } =12\sec ^{ 2 }{ \beta }$

Hence minimum value is 12 for $\sec { \beta } =1$

Calculus is not needed II

The answer is 12.

1 solution

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