Position of a Point (x₁,y₁) w.r.t. a Parabola y² =4ax

If $\left(a^2,a-2\right)$ be a point interior to the region of the parabola $y^2=2x$ bounded by the chord joining the points $\left(2,2\right)$ and $\left(8,-4\right)$, then find the set of all possible real values of $a$.

Equation of the parabola whose axis is parallel to $\mathrm{y}$-axis and which passes through the points $\left(1,0\right),\left(0,0\right)$ and $\left(-2,4\right)$ is :