Answer
(a) See the graph.
$k=1$:
$\frac{x^2}{17}+\frac{y^2}{1}=1~~$ (Red)
$k=2$:
$\frac{x^2}{20}+\frac{y^2}{4}=1~~$ (Green)
$k=4$:
$\frac{x^2}{32}+\frac{y^2}{16}=1~~$ (Blue)
$k=8$:
$\frac{x^2}{80}+\frac{y^2}{64}=1~~$ (Black)
(b)
Foci: $F(±4,0)$, no matter the value of $k$
Work Step by Step
(b)
Equation of an ellipse with center at $(h,k)$ (major axis is horizontal):
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$\frac{(x-h)^2}{16+k^2}+\frac{(y-k)^2}{k^2}=1$
So:
$a^2=16+k^2$
$b^2=k^2$
$a^2=b^2+c^2$
$c^2=a^2-b^2=16+k^2-k^2=16$
$c=4$
Foci: $F(±c,0)=F(±4,0)$
