Question

In triangle $ABC$, the line joining the circumcenter and orthocenter is parallet to side $AC$, then the value of $\tan A\tan C$ is equal to

• A. $\sqrt{3}$
• B. $3$
• C. $3\sqrt{3}$
• D. none of these

Answer 1

The correct option is B $3$

Distance of circumcentre from side AC is $R\cos B$
Distance orthocentre from side AC is $2R\cos A\cos C$
$\Rightarrow R\cos B=2R\cos A\cos C$
$\Rightarrow \cos (\pi -(A+C\left)\right)=2\cos A\cos C$
$\Rightarrow -\cos (A+C)=2\cos A\cos C$
$\Rightarrow \sin A\sin C=3\cos A\cos C$
$\Rightarrow \tan A\tan C=3$