Finding a distribution whose KL Divergence from a given distribution is a constant $\alpha$

Question

Consider P as a multinomial distribution over k variables. I would like to find a distribution Q, also a multinomial distribution over k variables such that KL Divergence between Q from P is a constant $\alpha$. In short what is the multinomial distribution Q over k variables, such that $D_{KL}(P||Q) = \alpha$. How can I solve for Q? More generally how can I solve for Q if both P and Q are from the same family of distributions and belong to the exponential family.