Q119 / 160 · WBJEE 2010 · Mathematics

In a triangle $\mathrm{PQR}, \angle \mathrm{R}=\pi / 2$. If $\tan \left(\frac{\mathrm{p}}{2}\right)$ and $\tan \left(\frac{\mathrm{Q}}{2}\right)$ are roots of $\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}=0$, where $\mathrm{a} \neq 0$, then which one is true ?

  • Correct answer
    A
    $c=a+b$
  • B
    $\mathrm{a}=\mathrm{b}+\mathrm{c}$
  • C
    $\mathrm{b}=\mathrm{a}+\mathrm{c}$
  • D
    $\mathrm{b}=\mathrm{c}$

Solution

Hints: $\frac{\mathrm{P}}{2}+\frac{\mathrm{Q}}{2}=\frac{\pi}{2}-\frac{\mathrm{P}}{2}=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$ $$ \begin{aligned} & \tan \left(\frac{\rho}{2}+\frac{Q}{2}\right)=1, \frac{-b / a}{1-c / a}=1 \Rightarrow \frac{-b}{a-c}=1 \\ & -b=a-c \Rightarrow a+b=c \end{aligned} $$

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

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