In this how they have got that term in the solution .

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Momentum $P$ is the same for both objects (conservation). Therefore $K\propto 1/m$.

Total KE is shared between the two objects in the ratio $\frac{1}{m_1} : \frac{1}{m_2}$. When something is shared in the ratio $a : b$ the objects have fractions $\frac{a}{a+b}$ and $\frac{b}{a+b}$ of the total.

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and then how you have written k$_1$