The amplitude of vibration of a particle is given by ${a_m} = ({a_0})/(a{\omega ^2} - b\omega + c);$ where ${a_0},a,b$ and $c$ are positive. The condition for a single resonant frequency is
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(a) For resonance amplitude must be maximum which is possible only when the denominator of expression is zero i.e. $a{\omega ^2} - b\omega + c = 0$
$ \Rightarrow \omega = \frac{{ + b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
For a single resonant frequency, $b^2 = 4ac.$
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