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(D) For capacitors in parallel,the equivalent capacitance is $C_{eq} = C_1 + C_2 + C_3$.Given $C_1 + C_2 + C_3 = 12$ $... (i)$Given $C_1 \cdot C_2 \cd

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$4, 2, 6$

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(D) For capacitors in parallel,the equivalent capacitance is $C_{eq} = C_1 + C_2 + C_3$.Given $C_1 + C_2 + C_3 = 12$ $... (i)$Given $C_1 \cdot C_2 \cdot C_3 = 48$ $... (ii)$Given $C_1 + C_2 = 6$ $... (iii)$Substituting $(iii)$ into $(i)$: $6 + C_3 = 12 \implies C_3 = 6$ units.Substituting $C_3 = 6$ into $(ii)$: $C_1 \cdot C_2 \cdot 6 = 48 \implies C_1 \cdot C_2 = 8$.We know $(C_1 - C_2)^2 = (C_1 + C_2)^2 - 4C_1 C_2$.$(C_1 - C_2)^2 = (6)^2 - 4(8) = 36 - 32 = 4$.Therefore,$C_1 - C_2 = 2$ $... (iv)$.Solving $(iii)$ and $(iv)$: $2C_1 = 8 \implies C_1 = 4$ and $C_2 = 2$.Thus,the capacitances are $C_1 = 4, C_2 = 2, C_3 = 6$.

Three capacitors $C_1$,$C_2$,and $C_3$ are connected in parallel,and their equivalent capacitance is $12$ units. Given $C_1 \cdot C_2 \cdot C_3 = 48$ units. When $C_1$ and $C_2$ are connected in parallel,their equivalent capacitance is $6$ units. Find the individual capacitances $C_1, C_2, C_3$.

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