The coefficient of coupling is 0.8 . If the mutual inductance is 8\...

The relation between mutual inductance $M$, self-inductances $L_1$ and $L_2$, and the coefficient of coupling $k$ is given by

$M = k \sqrt{L_1 L_2}$

Substituting the given values $M = 8\ \text{H}$ and $k = 0.8$, we get

$8 = 0.8 \sqrt{L_1 L_2}$

$\sqrt{L_1 L_2} = \dfrac{8}{0.8} = 10$

Squaring both sides, we obtain

$L_1 L_2 = 100$

Checking the given options, the product of $5\ \text{H}$ and $20\ \text{H}$ is $100$.

Therefore, the values of self-inductances are $5\ \text{H}$ and $20\ \text{H}$.

Answer: $5\ \text{H}$ and $20\ \text{H}$