Answer
(a) Vertices: $(±4,0)$
Foci: $(±2\sqrt {7},0)$
Asymptotes:
$y=±\frac{\sqrt 3}{2}x~~$ (Black lines)
(b) Length of the transverse axis:
$2a=8$
Work Step by Step
Hyperbola with horizontal transverse axis:
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$\frac{x^2}{16}-\frac{y^2}{12}=1$
$a^2=16$
$a=4$
Vertices: $(±a,0)=(±4,0)$
$b^2=12$
$b=2\sqrt {3}$
$c^2=a^2+b^2=16+12=28$
$c=2\sqrt {7}$
Foci: $(±c,0)=(±2\sqrt {7},0)$
Asymptotes:
$y=±\frac{b}{a}x$
$y=±\frac{2\sqrt {3}}{4}x=±\frac{\sqrt 3}{2}x$
Length of the transverse axis:
$2a=8$
