Question

In triangle $ABC$, line joining the circumcentre and orthocentre is parallel to side $AC$ then 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 $3$

Consider $△ABC$ where $C$ is the circumcentre and $O$ is the orthocentre.

Extend $C$ and $O$ to meet $AC$ in $E$ and $F$ respectively.

We know that

$\Rightarrow CE=R\cos B$

$\Rightarrow OF=2R\cos A\cos C$

$\Rightarrow CE=OF$

$\Rightarrow R\cos B=2R\cos A\cos C$

$\Rightarrow \cos B=2\cos A\cos C$

$\Rightarrow \cos (\pi -(A+C\left)\right)=2\cos A\cos C$

$\Rightarrow -\cos (A+C)=2\cos A\cos C$

$\Rightarrow -(\cos A\cos C-\sin A\sin C)=2\cos A\cos C$

$\Rightarrow \sin A\sin C=3\cos A\cos C$

$\Rightarrow \tan A\tan C=3$