Step 1:
$x^2+4y^2-8x-16y-68=0$
$x^2-8x+4y^2-16y=68$
$(x^2-8x+16)+4(y^2-4y+4)=68+16+16$
$(x-4)^2+4(y-2)^2=100$
The above equation is divided by $100$
Step 2:
$\large\frac{(x-4)^2}{100}+\frac{(y-2)^2}{25}$$=1$
Shifting the origin to $(4,2)$ by translation of axes.
$X=x-4$ (or $x=X+4)$
$Y=y-2$ (or $Y=y+2$)
$\large\frac{X^2}{100}+\frac{Y^2}{25}$$=1$
$a^2=100,b^2=25$
$\Rightarrow a=10,b=5$.
Step 3:
The major axes is the $X$-axes.(i.e) $Y=0$ or $y-2=0$
Step 4:
$XY$-axes :
Eccentricity $e=1-\sqrt{\large\frac{b^2}{a^2}}$
$\Rightarrow \sqrt{1-\large\frac{25}{100}}=\sqrt{\frac{75}{100}}=\large\frac{\sqrt3}{2}$
Centre : $(0,0)$
Foci : $(\pm ae,0)$
$ae=10\times \large\frac{\sqrt 3}{2}=$$5\sqrt 3$
$\Rightarrow (\pm 5\sqrt 3,0)$
Vertices : $(\pm a,0)$
$\Rightarrow (\pm 10,0)$
Step 5:
$xy$-axes :
Centre : $(4,2)$
Foci : $(\pm 5\sqrt 3+4,2)$
Vertices : $(\pm 10+4,2)$
$\Rightarrow (14,2),(-6,2)$