Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices: (-2,1),(2,1);
passes through the point (5,4)
The equation of a hyperbola is given by $\displaystyle \frac{(x-h)^2}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$, where $\displaystyle (h,k)$ is the centre of the hyperbola.
For your question, the centre of the hyperbola is $\displaystyle (0,1)$, and $\displaystyle a=2$ since $\displaystyle a$ is the distance from the vertices to the centre.
Thus, $\displaystyle \frac{(x-0)^2}{2^{2}}-\frac{(y-1)^{2}}{b^{2}}=1$
Since the curve passes through $\displaystyle (5,4)$, you can find $\displaystyle b^{2}$ by substitution.