Select the necessarily true statements involving conditional expectation and the joint Gaussian distribution of $X$ and $Y$

Question

Question: $X$ and $Y$ have a joint distribution which is Gaussian. The means of $X$ and $Y$ are equal to zero, and their variances strictly positive. The correlation between $X$ and $Y$ is strictly between −1 and 1.

Select all the necessarily true statements:

(a) If $P(E[X|Y]=0)=1$ then $X$ and $Y$ are independent.

(b) $\frac{E[X|Y]E[Y|X]}{XY} \leq 1$

(c) $ E[eX|Y]=e^{aY}$ for some $a \in \mathbb{R}$.

(d) The conditional distribution of $X+Y$ given $X=x$ is equal to the distribution of $x+Y$.

I am able to answer (d) and believe it is true. However I am struggling to determine if (a),(b) and (c) are true or false.

Any help?