Answer
$(0, -a)$, $(0, a)$, $c=\sqrt {a^2−b^2}$, $(0,-5)$, $(0,5)$, $(0,-3)$, $(0,3)$.
Work Step by Step
1. Recall the definition and the general equation of an ellipse, with the given equation $\frac{x^2}{b^2} + \frac{y^2}{a^2}=1$ where $a\gt b\gt0$, we can identify its vertices as $(0, -a)$ and $(0, a)$. (Please note that in this case the vertices are on the y-axis)
2. The foci is given as $(0,\pm c)$, with the relationship formula $c^2=a^2−b^2$, we have $c=\sqrt {a^2−b^2}$. (Please note that in this case the foci are on the y-axis)
3. In the example of $\frac{x^2}{4^2} + \frac{y^2}{5^2}=1$, we have $a=5,b=4$ and $c=\sqrt {5^2−4^2}=3$, so the vertices are $(0,-5)$ and $(0,5)$ and the foci are $(0,-3)$ and $(0,3)$.
