Resonant frequency of RLC circuit if for two values of omega we have same max amplitude current

Question

Original question:

An alternating current is flowing through a series LCR circuit. It is found that the current reaches a value of 1 mA at both 200 Hz and 800 Hz frequency. What is the Resonance frequency of the circuit?

My method was that according to given conditions for two frequencies w and w' we have \$\left(\omega L-\frac{1}{\omega c}\right)^{2}=\left(\omega^{\prime} L-\frac{1}{\omega^{\prime} c}\right)^{2}\$ so from here we get that either \$\omega+\omega' = 1/\sqrt{LC} = 1000\;\text{Hz}\$, or \$\omega-\omega' = 1/\sqrt{LC} = 600\;\text{Hz}\$, so resonant frequency is either of the two but given is that resonant frequency is \$\sqrt{\omega\omega'} = 400\;\text{Hz}\$, why is that so ?

Answer 1

Some guidance: You need to find the LC because the resonant frequency is 1/(2pi*sqrt(LC))

Your equation \$\left(\omega L-\frac{1}{\omega c}\right)^{2}=\left(\omega^{\prime} L-\frac{1}{\omega^{\prime} c}\right)^{2}\$ has demolished a part of the information. Your equation is true as a consequence but it does not reversely imply the right solution for LC, it gives 2 possible values and only one of them is the right LC.

BTW. I do not claim you have found any of them. Seems like you have sometimes forgotten to let number 2pi hang around when one talks of hertzes, not of radians per a second.

A way to keep the things in hands is to rethink what we actually know and derive the right LC from it with equivalent equation transformations, not by writing a consequence which allows also wrong solutions.

We know that complex impedance R + jX has the same absolute value at 200Hz and at 800Hz and only the imaginary part X depends on frequency. The only possibility to get it is that X has exactly opposite values at 200Hz and 800Hz.

So: Write it as your starting equation and get the one and only right solution for LC.