Question

In a $\Delta ABC,\tan A$ and $\tan B$ are the roots of $pq(x^2+1)=r^{2}x$. Then, $\Delta ABC$ is

• A. A right angled triangle
• B. An acute angled triangle
• C. An obtuse angled triangle
• D. An equilateral triangle

Answer 1

The correct option is A A right angled triangle

Given equation can be written as
$x^2-\frac{r^2}{pq}x+1=0$

$\therefore \tan A+\tan B=\frac{r^2}{pq}$

and $\tan A\tan B=1$

We know, $A+B+C={180}^{}$

$\Rightarrow (A+B)={180}^{}-C$

$\Rightarrow \tan (A+B)=\tan ({180}^{}-C)$

$\Rightarrow \frac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C$

$\Rightarrow \frac{\frac{r^2}{pq}}{1-1}=-\tan C$

$\Rightarrow \infty =-\tan C\Rightarrow C={90}^{}$

Hence, $\Delta ABC$ is a right angled triangle.