Answer
$\dfrac{5(y-4)^{2}}{16}-\dfrac{5(x-2)^{2}}{64}=1$; equation of a hyperbola.
Work Step by Step
From asymptotes we have $\frac{a}{b}=\frac{1}{2}$
or, $2a=b$
and $a^2+b^2=c^2$ or, $a^2+(2a)^2=4^2$
$a=\frac{4}{\sqrt 5}$ and $b=2a=2(\frac{4}{\sqrt 5})=\frac{8}{\sqrt 5}$
Hence, $\dfrac{5(y-4)^{2}}{16}-\dfrac{5(x-2)^{2}}{64}=1$; equation of a hyperbola.
