Question

If the median of $△ABC$ through $A$ is perpendicular to $AB$, then

• A. $\tan A+\tan B=0$
• B. $2\tan A+\tan B=0$
• C. $\tan A+2\tan B=0$
• D. none of these

Answer 1

Answer: (C)

$\tan A+2\tan B=0$

Applying $m-n$ co-tangent theorem, to $△ABC$ we have
$(\frac{a}{2}+\frac{a}{2})\cot (90+B)$
$=\frac{a}{2}\cot \frac{\pi }{2}-\frac{a}{2}\cot (A-\frac{\pi }{2})$
$\Rightarrow -a\tan B=0+\frac{a}{2}\tan A$
$\Rightarrow -a\tan B=\frac{a}{2}\tan A$
$\therefore \tan A+2\tan B=0$