Answer
$\dfrac{x^2}{7}+\dfrac{y^2}{16}=1$
Work Step by Step
The standard equation of an ellipse having a vertex and focus on the $y$-axis is:
$\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1$
Determine $h,k$ using the center's coordinates:
$(h,k)=(0,0)$
$h=0$
$k=0$
Determine $a$ using the vertex:
$(0,4)=(h,k+a)$
$k+a=4$
$a=4$
Determine $c$ using the focus:
$(h,k+c)=(0,3)$
$k+c=3$
$c=3$
Determine $b^2$:
$c^2=a^2-b^2$
$b^2=a^2-c^2$
$b^2=4^2-3^2$
$b^2=7$
The ellipse's equation is:
$\dfrac{x^2}{7}+\dfrac{y^2}{4^2}=1$
$\dfrac{x^2}{7}+\dfrac{y^2}{16}=1$
