Question

In a triangle PQR, $\angle R=\frac{\pi }{2}$. If $tan\left(\begin{array}{c}\frac{P}{2}\end{array}\right)$ and $tan\left(\begin{array}{c}\frac{Q}{2}\end{array}\right)$ are roots of $a{x}^2+bx+c=0$, where $a\ne =0$, then which one is true?

• A. $c=a+b$
• B. $a=b+c$
• C. $b=a+c$
• D. $b=c$

Answer 1

The correct option is A $c=a+b$

$\frac{P}{2}+\frac{Q}{2}=\frac{\pi }{2}-\frac{P}{2}=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}$
$tan\left(\begin{array}{c}\frac{P}{2}+\frac{Q}{2}\end{array}\right)=1\Rightarrow \frac{\frac{-b}{a}}{1-\frac{c}{a}}=1$
$\Rightarrow \frac{-b}{a-c}=1$
$\Rightarrow -b=a-c$
$\Rightarrow a+b=c$