Question

Three vertical poles of heights $h_1,h_2$ and $h_3$ at the vertices $A,B$ and $C$ of a $△ABC$ subtend angles $\alpha ,\beta ,\gamma$ respectively at the circumcentre of the triangle. If $\cot \alpha ,\cot \beta ,\cot \gamma$ are in AP, then $h_1,h_2,h_3$ are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A AP

Let $R$ be the circumradius and $O$ be the circumcentre of $△ABC$
In $△OA{P}_1$ we have
$\tan \alpha =\frac{A{P}_1}{OA}\Rightarrow h_1=R\cot \alpha$
Similarly $h_2=R\cot \beta ;h_3=R\cot \gamma$
Since $\cot \alpha ,\cot \beta ,\cot \gamma$ are in AP.
Therefore, $2\cot \beta =\cot \alpha +\cot \gamma$
$\Rightarrow 2R\cot \beta =R\cot \alpha +R\cot \gamma$
$\Rightarrow 2h_2=h_1+h_3$
$\therefore$ $h_1,h_2,h_3$ are in AP.