Answer
Center: $(6,-7)$
Foci: $(6,-7+2\sqrt 3)~~and~~(6,-7-2\sqrt 3)$
Vertices: $(6,-2)~~and~~(6,-11)$
Work Step by Step
The standard form of the equation of the elipse when the major axis is:
- horizontal: $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ in which $(h,k)$ is the center and $2a$ is the major axis length and $2b$ is the minor axis length.
- vertical: $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$ in which $(h,k)$ is the center and $2a$ is the major axis length and $2b$ is the minor axis length.
$\frac{(x-6)^2}{4}+\frac{(y+7)^2}{16}=1$
$\frac{(x-6)^2}{2^2}+\frac{(y+7)^2}{4^2}=1$
$\frac{(x-6)^2}{2^2}+\frac{[y-(-7)]^2}{4^2}=1$
$a=4,~~b=2$
$c^2=a^2-b^2=16-4=12$
$c=2\sqrt 3$
Center: $(6,-7)$
The major axis is vertical:
- the foci: $(6,-7+2\sqrt 3)~~and~~(6,-7-2\sqrt 3)$
- the vertices: $(6,-6+4)=(6,-2)~~and~~(6,-7-4)=(6,-11)$
