First slide

Theory of expressions

Question

In a triangle PQR, $∠ R = π / 2 .$ If $\tan (P / 2)$ and $\tan (Q / 2)$ are the roots of the equation $a x^{2} + b x + c = 0 (a \neq 0)$
then

Moderate

A

$a + b = c$
B

$a + b = 0$
C

$a + c = b$
D

$b = c$

Solution

It is given that $\tan \frac{P}{2}$ and $\tan \frac{Q}{2}$ are roots of
$a x^{2} + b c + c = 0$
$∴ \tan \frac{P}{2} + \tan \frac{Q}{2} = - \frac{b}{a}$ and $\tan \frac{P}{2} \tan \frac{Q}{2} = \frac{c}{a}$
We have
$R = \frac{π}{2}$ and $P + Q + R = π \Rightarrow P + Q = \frac{π}{2} \Rightarrow \frac{P}{2} + \frac{Q}{2} = \frac{π}{4}$
$\begin{array}{l} ∴ \tan (\frac{P}{2} + \frac{Q}{2}) = \tan \frac{π}{4} \\ \to \frac{\tan P / 2 + \tan Q / 2}{1 - \tan P / 2 \tan Q / 2} = 1 \\ \Rightarrow \frac{- \frac{b}{a}}{1 - \frac{c}{a}} = 1 \end{array}$
$\Rightarrow 1 - \frac{c}{a} = - \frac{b}{a} \Rightarrow a - c = - b \Rightarrow a + b = c$

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