Two resistors when connected together can produce maximum 40 ohm and minimum 10 ohm resistance. Find the resistance of each resistor.

Explanation

Let we have two resistors of - resistances `R_1 and R_2`
`R_("maxi") = 40 Omega`
`R_("mini") = 10Omega`
These resistors produce maximum resistance when connected in series.
`:. R_1 + R_2 = 40 " "....(1)`
These resistors produce minimum resistance when connected in parallel.
`:. 1/R_1+1/R_2=1/10`
`(R_1+R_2)/(R_1R_2)=1/10`
`40/(R_1R_2)=1/10`
`[ :. R_(1)+R_2=40Omega]`
`R_1R_2 = 400`
We know `(R_1-R_2) = (R_1+R_2)^(2) - 4R_(1)R_2`
`(R_1-R_2)^2=(40)^2-4(400)`
`= 1600-1600 =0`
`implies R_(1)-R_2=0`
`implies R_1 =R_2`
From eqn. (1) `R_1+R_2 =40`
`R_1=20Omega`
`:. R_1 = R_2`
`:. R_(2) = 20Omega`