Question

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

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

Answer 1

The correct option is A

A right-angled triangle

Explanation For The Correct Option:

Determining $∆ABC$:

The given equation is

$pq(x^2+1)=r^{2}x$

This equation can be written as

$pq{x}^2-r^{2}x+pq=0$

This is a quadratic equation then

Sum of roots of this equation is $=\frac{-\left(-r^2\right)}{pq}$

As given $\tan A$and $\tan B$ are the roots of this equation

$$\begin{gathered} \Rightarrow \tan A+\tan B=\frac{r^2}{pq}\dots \dots \left(1\right) \\ \text{We know that} \\ \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} \\ \text{Substituting from equation}\left(1\right) \\ \Rightarrow \tan (A+B)=\frac{\frac{r^2}{pq}}{1-1};\left[\because \tan A\tan B=1\right] \\ =\infty \\ \Rightarrow \tan (A+B)=\tan 90° \\ \Rightarrow A+B=90° \\ \because A+B+C=180° \\ \Rightarrow C=90° \end{gathered}$$

As one of the angle in a triangle is right angled then it is a right-angled triangle.

Hence, the correct answer is option (A).