Answer
$\dfrac{x^2}{9}+\dfrac{y^2}{25}=1$
See graph
Work Step by Step
We are given the ellipse:
Center: $(0,0)$
Focus: $(0,-4)$
Vertex: $(0,5)$
Because the $x$-coordinates of the vertex and focus are the same, the ellipse has the equation:
$\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1$
Determine $h,k$ using the center:
$(h,k)=(0,0$
$h=0$
$k=0$
Determine $a$ using the vertex:
$(h,k+a)=(0,5)$
$(0,0+a)=(0,5)$
$(0,a)=(0,5)$
$a=5$
Determine $c$ using the focus:
$(h,k-c)=(0,-4)$
$(0,0-c)=(0,-4)$
$(0,-c)=(0,-4)$
$c=4$
Determine $b$:
$a^2=b^2+c^2$
$b^2=a^2-c^2$
$b^2=5^2-4^2$
$b^2=9$
$b=\sqrt 9=3$
The equation of the ellipse is:
$\dfrac{x^2}{3^2}+\dfrac{y^2}{5^2}=1$
$\dfrac{x^2}{9}+\dfrac{y^2}{25}=1$
The center is:
$(h,k)=(0,0)$
Graph the ellipse:
