Question

In a $△ABC$, $\tan B+\tan C=5$ and $\tan A\tan C=3$, then which of the following is/are correct?

• A. $△ABC$ is an acute angle triangle.
• B. $△ABC$ is an obtuse angle triangle.
• C. Sum of all possible value of $\tan A$ is $10.$
• D. Sum of all possible value of $\tan A$ is $9.$

Answer 1

Answer: (A), (C)

The correct options are
A $△ABC$ is an acute angle triangle.
C Sum of all possible value of $\tan A$ is $10.$

As $A+B+C=\pi$, so
$\tan A+\tan B+\tan C=\tan A\tan B\tan C\Rightarrow \tan A+5=3\tan B\Rightarrow \tan A+5=3(5-\tan C)\Rightarrow \tan A+5=3(5-\frac{3}{\tan A})\Rightarrow {\tan }^{2}A+5\tan A=15\tan A-9\Rightarrow {\tan }^{2}A-10\tan A+9=0\Rightarrow (\tan A-1)(\tan A-9)=0\therefore \tan A=1,9$
Sum of all possible value of $\tan A$ is $10.$
So,
$\tan C=3,\frac{1}{3}\tan B=2,\frac{14}{3}$
Therefore, the triangle is acute angle.