A Constant Slope Combination on a Line

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has a vertex $A(2,0)$ and eccentricity $\dfrac{\sqrt 2}2$. The line $y=k(x-1)$ meets $C$ at two distinct points $M$ and $N$. Through $M$, the line of slope $1$ meets the vertical line $x=t$ at a point $P$.

1. Find the equation of the ellipse $C$. 2. Let $k_1$ be the slope of $PN$ and $k_2$ the slope of $PA$. Does there exist a real number $t$ such that $4k_2-3k_1$ is constant? If so, find $t$; if not, explain why.