By Cholesky decomposition, you can express the normal random variables $X$ and $Y$ in the form \begin{align*} Y &= E(Y) + \sqrt{Var(Y)}\, \xi,\\ X &= E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right), \end{align*} where $\rho = \frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ is the correlation, $\xi$ and $\eta$ are two independent standard normal random variables.

Then, \begin{align*} E(X \mid Y) &= E\left(E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right) \mid \xi \right)\\ &=E(X) + \rho \sqrt{Var(X)}\xi\\ &=E(X) + \frac{Cov(X, Y)}{\sqrt{Var(Y)}}\xi\\ &=E(X) + \frac{Cov(X, Y)}{Var(Y)}\big(Y-E(Y) \big). \end{align*} The computation for $Var(X\mid Y)$ is similar, and I will leave for you to figure it out.

By Cholesky decomposition, you can express the normal random variables $X$ and $Y$ in the form \begin{align*} Y &= E(Y) + \sqrt{Var(Y)}\, \xi,\\ X &= E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right), \end{align*} where $\rho = \frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ is the correlation, $\xi$ and $\eta$ are two independent standard normal random variables.

Then, \begin{align*} E(X \mid Y) &= E\left(E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right) \mid \xi \right)\\ &=E(X) + \rho \sqrt{Var(X)}\xi\\ &=E(X) + \frac{Cov(X, Y)}{\sqrt{Var(Y)}}\xi\\ &=E(X) + \frac{Cov(X, Y)}{Var(Y)}\big(Y-E(Y) \big). \end{align*} The computation for $Var(X\mid Y)$ is similar, specifically, \begin{align*} Var(X \mid Y) &=E\left((X-E(X\mid Y))^2\mid Y \right)\\ &=E\left( (X-E(X\mid Y))^2\mid \xi\right)\\ &=E\left(Var(X)(1-\rho^2) \eta^2 \mid \xi\right)\\ &=E\left(Var(X)(1-\rho^2) \eta^2\right)\\ &=Var(X)(1-\rho^2). \end{align*}

By Cholesky decomposition, you can express the normal random variables $X$ and $Y$ in the form \begin{align*} X &= E(X) + \sqrt{Var(X)}\, \xi,\\ Y &= E(Y) + \sqrt{Var(Y)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right), \end{align*} where $\rho = \frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ is the correlation, $\xi$ and $\eta$ are two independent standard normal random variables.

By Cholesky decomposition, you can express the normal random variables $X$ and $Y$ in the form \begin{align*} Y &= E(Y) + \sqrt{Var(Y)}\, \xi,\\ X &= E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right), \end{align*} where $\rho = \frac{Cov(X, Y)}{\sqrt{Var(X)Var(Y)}}$ is the correlation, $\xi$ and $\eta$ are two independent standard normal random variables.

Then, \begin{align*} E(X \mid Y) &= E\left(E(X) + \sqrt{Var(X)}\left(\rho \xi+\sqrt{1-\rho^2} \eta\right) \mid \xi \right)\\ &=E(X) + \rho \sqrt{Var(X)}\xi\\ &=E(X) + \frac{Cov(X, Y)}{\sqrt{Var(Y)}}\xi\\ &=E(X) + \frac{Cov(X, Y)}{Var(Y)}\big(Y-E(Y) \big). \end{align*} The computation for $Var(X\mid Y)$ is similar, and I will leave for you to figure it out.