The ellipse E:(x^2/(9)+(y^2/(5)=1 and the parabola P:y^2=8x are drawn in the same plane.

Substituting $y^2=8x$ in $E$ gives $5x^2+72x-45=0$, i.e. $x=\frac35$ or $x=-15$. Only $x=\frac35$ is admissible (since $x\ge0$ and $|x|\le3$), with $y=\pm\frac{2\sqrt{30}}5$, so there are exactly two points and (A) is true. The slopes are $\frac4y$ (for $P$) and $-\frac{5x}{9y}$ (for $E$); their product is $-\frac{20x}{9y^2}=-\frac{20(3/5)}{9(24/5)}=-\frac{5}{18}\ne-1$, so (B) is false. The focus of $P$ is $(2,0)$, also a focus of $E$ ($c^2=4$); the distance from it is $x+2=\frac{13}{5}$, so the sum $2a=6$ gives $6-\frac{13}5=\frac{17}5$ from $(-2,0)$ and (C) is true. The tangent to $E$ cuts the axis at $\frac{9}{x_0}=15$, not $9$, so (D) is false.