Parabola question bank (Maths)

Let $A,B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of latus rectum of $C_o$ is ______

Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of $P$ and $E$, be their latus rectums. If the eccentricity of $E$ is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.