conditionnal moments of a normal distribution

[finanmath]

1. The problem statement, all variables and given/known data

We have Vx,Vy following a Normal standardized distribution
from which we construct the following correlated variables: X, Y.
We consider the events such that x(belong to)A, with 0 < Pr[x(belong to) A] <1.
We want to compute V(Y|X E A), Cov(X,Y|X E A) in order to compute the correlation over the events A ?

2. Relevant equations

X=Mux+Vx*Sx,
Y=Muy+ Sy*(rho*Vx+Vy*(1-rho^2)^0.5 )

3. The attempt at a solution

I already computed the conditionnal covariance and end up with :

COV(X,Y|X E A)=rho*Sy/Sx*V(X|X E A)

[finanmath]

Please Help me ! I am feeling very lost in this exercise and can't do it without someone's help. Don't hesitate to ask me any questions.

[finanmath]

In fact I have started this way for the conditionnal variance, but I m not sure if it s right:
E=belong to (logic operator)

V(Y|x E A)=V(Muy + Sy*(rho*Vx+Vy*(1-rho^2)^0.5 )|x E A)=V(Sy*(rho*Vx+Vy*(1-rho^2)^0.5 )|x E A)=Sy^2*rho^2*V(Vx/X E A)+ Sy^2*(1-rho^2)*V(Vy|x E A)
there is no covariance term in the last equality but my issue is to comput the two remaining variance. I know that the unconditionnal variance of Vx and Vy is 1(standardized normal).
Though if u can help me I ll be grateful.