Speed of hypothetical planets

Question

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

Answer 1

$K \propto 1/m$

$K_1 = \frac{1/m_1}{1/m_1 + 1/m_2} \times K_{tot}= \frac{m_2}{m_1 + m_2} K_{tot}$.

Comments

why $K \propto 1/m$
and then how you have written k$_1$

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$K=\frac{P^2}{2m}$  
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.