Answer
See graph
Foci: $(-\sqrt{10},0),(\sqrt{10},0)$
Work Step by Step
We are given the hyperbola:
$\dfrac{x^2}{9}-y^2=1$
Bring the equation to the standard form:
$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$
$\dfrac{x^2}{9}-\dfrac{y^2}{1}=1$
The transverse axis is parallel to the $x$-axis.
Determine $h,k,a,b,c$:
$h=0$
$k=0$
$a^2=9\Rightarrow a=\sqrt 9=3$
$b^2=1\Rightarrow b=\sqrt {1}=1$
$c^2=a^2+b^2$
$c^2=9+1$
$c^2=10$
$c=\sqrt{10}$
The centre of the hyperbola is:
$(h,k)=(0,0)$
Determine the coordinates of the vertices:
$(h-a,k)=(0-3,0)=(-3,0)$
$(h+a,k)=(0+3,0)=(3,0)$
Determine the coordinates of the foci:
$(h-c,k)=(0-\sqrt{10},0)=(-\sqrt{10},0)$
$(h+c,k)=(0+\sqrt{10},0)=(\sqrt{10},0)$
Determine the asymptotes:
$y-k=\pm\dfrac{b}{a}(x-h)$
$y-0=\pm\dfrac{1}{3}(x-0)$
$y=\pm\dfrac{1}{3}x$
Graph the hyperbola:
