Consider two quadratic equations \( a_{1} x^{2}+b_{1} x+c_{1}=0 \) and \( a_{2} x^{2}+b_{2} x+c_{2}=0 \). Let \( \alpha \) be the root that is common in the quadratic equation.
\( \mathrm{P} \)
Then, \( a_{1} \alpha^{2}+b_{1} \alpha+c_{1}=0 \) and \( a_{2} \alpha^{2}+b_{2} \alpha+c_{2}=0 \)
Using cramer's rule,
\[
\frac{\alpha^{2}}{\left|\begin{array}{ll}
-c_{1} & b_{1} \\
-c_{2} & b_{2}
\end{array}\right|}=\frac{\alpha}{\left|\begin{array}{ll}
a_{1} & -c_{1} \\
a_{2} & -c_{2}
\end{array}\right|}=\frac{1}{\left|\begin{array}{ll}
a_{1} & b_{1} \\
a_{2} & b_{2}
\end{array}\right|}
\]
The condition for only one common root is \( \left(c_{1} a_{2}-c_{2} a_{1}\right)^{2}=\left(b_{1} c_{2}-b_{2} c_{1}\right)\left(a_{1} b_{2}-a_{2} b_{1}\right) \) If both roots are common, then the condition is \( \left(a_{1} / a_{2}\right)=\left(b_{1} / b_{2}\right)=\left(c_{1} / c_{2}\right) \).
If \( a, b, c \) belong to \( R \) and equations \( a x^{2}+b x+c=0 \) and \( x^{2}+2 x+9=0 \) have a common root, than \( a: b: c= \) ?
(1) \( 1: 2: 3 \)
(2) \( 2: 1: 9 \)
(3) \( 1: 2: 9 \)
(4) \( 9: 2: 1 \)
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