Let $X$ and $Y$ independent random variables distributed as $N(0,1)$. For any $a\in [-1,1]$, and any positive $t\in\mathbb{R}$ how to find an upper bound of the joint probability $P(aX+\sqrt{1-a^2}Y>t, X>t)$? Here $t$ can be thought of a large number so that we are interested in the upper tail and using standard normal tail bound $P(X>t)\leq e^{-t^2/2}$.
My primary interest is to bound to the quantity $P(aX+\sqrt{1-a^2}Y>t | X>t)$, which can be written as $\frac{P(aX+\sqrt{1-a^2}Y>t, X>t)}{P(X>t)}$.
For the conditional probability, consider three extreme cases:
i) $a=1$, then $P(X>t | X>t)=1$
ii) $a=0$, then $P(Y>t|X>t)=P(Y>t)\leq e^{-t^2/2}$ (since $X$ and $Y$ are independent and using tail bound of $Y$).
iii) $a=-1$, then $P(-X>t|X>t)=0$.
For any other values of $a$, how to find an upper bound of the conditional probability as a function of $a$?