An `A.C` circuit contains a resistor `'R'` an inductor `'L'` and a capacitor `'C'` connected in series. When it is connected to an `A.C` generator of fixed output voltage and varialbe frequency, the current in the circuit is found to be leading the applied voltage `(pi)/(4)` read, when the frequency is `f_(1)`. when the frequency of the generator increased to `f_(2)` the current is found to be lagging behind the applied voltage by `(pi)/(4)` rad. The resonant frequency of the circuit is

Options

A. `(f_(1) f_(2))/(f_(1) + f_(2))`
B. `(f_(1) + f_(2))/(2)`
C. `(2 f_(1) f_(2))/(f_(1) + f_(2))`
D. `sqrt(f_(1) f_(2))`

Answer

D

Explanation

`"tan" (pi)/(4) = ((1)/(2 pi f_(1) C) - 2 pi f_(1) L)/(R )` and `"tan" (pi)/(4) = (2 pi f_(2) L - (1)/(2 pi f_(2) C))/(R )`
Resonant frequency `= (1)/(2 pi sqrt(LC))`