Q74 / 200 · WBJEE 2014 · Mathematics

In a $\triangle A B C$, tan $A$ and $\tan B$ are the roota $p q\left(x^{2}+1\right)=r^{2} x$ Then, $\Delta A B C$ is

  • Correct answer
    A
    a right angled triangle
  • B
    an acute angled triangle
  • C
    an obtuse angled triangle
  • D
    an equilateral triangle

Solution

Given equation can be rewritten as $x^{2}-\frac{r^{2}}{p q} x+1=0$ $\therefore \quad \tan A+\tan B=\frac{r^{2}}{p q}$ and $\tan A \tan B=1$ We know, $A+B+C=180^{\circ}$ $\Rightarrow \quad(A+B)=180^{\circ}-C$ $\Rightarrow \quad \tan (A+B)=\tan \left(180^{\circ}-C\right)$ $\Rightarrow \quad \frac{\tan A+\tan B}{1-\tan A \tan B}=-\tan C$ $\Rightarrow \quad \frac{r^{2} / p q}{1-1}=-\tan C$ $\Rightarrow \quad \infty=-\tan C \Rightarrow C=90^{\circ}$ Hence, $\triangle A B C$ is a right angled triangle.

Hence, the correct answer is option: 'A'.

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