# When two resistors of resistances R_1 and R_2 are connected in parallel, the net resistance is 3 Omega. When connected in series, its value is 16

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When two resistors of resistances R_1 and R_2 are connected in parallel, the net resistance is 3 Omega. When connected in series, its value is 16 Omega. Calculate the values of R_1 and R_2.

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When R_1 and R_2 are connected in parallel, net resistance (R_p) is given by
(1)/(R_p) = (1)/(R_1) + (1)/(R_2) = (R_1 + R_2)/(R_1 R_2)
or R_p = (R_1 R_2)/(R_1 + R_2) = 3 …(i)
When R_1 and R_2 are connected in series, net resistance (R_s) given by
R_s = R_1 + R_2 = 16 ...(ii)
From eqns. (i) and (ii), (R_1 R_2)/(16) = 3 or R_1 R_2 = 48
or R_1(16 - R_1) = 48 ("as" R_1 + R_2 = 16, R_2 = 16 - R_1)
or 16 R_1 - R_1^2 = 48
or R_1^2 - 16 R_1 + 48 = 0
or R_1^2 - 4 R_1 - 12 R_1 + 48 = 0
or R_1(R_1 - 4) - 12 (R_1 - 4) = 0 or (R_1 - 12)(R_1 - 4) = 0
From eqn. (i), R_2 = 4 Omega or 12 Omega
Therefore, the ressistance of two resistors are 4 Omega and 12 Omega`.