Q73 / 170 · Manipal (MET) 2019 · Mathematics
If the equation $x^2+p x+q=0$ and $x^2+q x+p=0$ have a common root then $p+q+1$ is equal to
- Correct answerA$0$
- B$1$
- C$2$
- D$-1$
Solution
Given equations are
$x^2+p x+q=0$ ...(i)
$x^2+q x+p=0$ ...(ii)
Let $\alpha$ be the common root, then
$\begin{aligned}
& \alpha^2+p \alpha+q=0 \text { and } \alpha^2+q \alpha+p=0 \\
& \Rightarrow \quad \frac{\alpha^2}{p^2-q^2}=\frac{\alpha}{q-p}=\frac{1}{q-p}
\end{aligned}$
$\Rightarrow \quad \alpha^2=\frac{(p-q)(p+q)}{-(p-q)}=-(p+q) \ldots$...(iii)
and $\quad \alpha=\frac{q-p}{q-p}=1$ ...(iv)
$\therefore$ From Eq. (iii)
$\begin{aligned} & & \alpha^2 & =-p+q \\ \Rightarrow & & (1)^2 & =-(p+q) \quad[\text { from Eq. (iv) }] \\ \Rightarrow & & p+q & =-1 \Rightarrow p+q+1=0\end{aligned}$
Hence, the correct answer is option: 'A'.