How to approximate a Mixture distribution by a single distribution

Question

Suppose there is a mixture of two Gaussian distributions $G = \alpha N_1 + (1-\alpha)N_2$, where $N_1,N_2$ are known Gaussian distributions, $\alpha$ is the ratio rate. I have two questions:

• How to find a gaussian distribution $N_{approx}$ which can minimum $KL-divergence(N_{approx}||G)$

• Can I get a new gaussian distribution $N_{new}$ from $N_1,N2$, that $N_{new}$ has a smaller standard deviation than $N_1,N_2$?

Thanks.