To find the standard form of the hyperbola, let's break down the information provided:

• Vertices: $(0, 2)$ and $(0, -2)$
• Foci: $(0, 4)$ and $(0, -4)$

Step 1: Identify the orientation of the hyperbola

The vertices and foci are located on the y-axis. This suggests that the hyperbola is vertical, which implies that the equation of the hyperbola takes the form:

$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Step 2: Calculate the distance from the center to the vertices

The distance from the center (which is at the origin, $(0,0)$) to the vertices is $a$, the semi-major axis length. The vertices are $(0, 2)$ and $(0, -2)$, so:

$a = 2$

Thus, $a^2 = 4$.

Step 3: Calculate the distance from the center to the foci

The distance from the center to the foci is $c$, the distance from the center to the focus point. The foci are $(0, 4)$ and $(0, -4)$, so:

$c = 4$

Step 4: Use the relationship between $a$, $b$, and $c$

For a hyperbola, the relationship between $a$, $b$, and $c$ is given by:

$c^2 = a^2 + b^2$

Substituting $a^2 = 4$ and $c^2 = 16$ into this equation:

$16 = 4 + b^2$

Solve for $b^2$:

$b^2 = 12$

Step 5: Write the standard form of the hyperbola

Now that we have $a^2 = 4$ and $b^2 = 12$, we can substitute these values into the standard form equation:

$\frac{y^2}{4} - \frac{x^2}{12} = 1$

This is the standard form of the given hyperbola.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the general form of a horizontal hyperbola?
2. How do you find the asymptotes of a hyperbola?
3. Can you derive the equation of a hyperbola centered at a different point than the origin?
4. What are the key differences between ellipses and hyperbolas?
5. How would the equation change if the foci were along the x-axis?

Tip: In hyperbolas, the relationship $c^2 = a^2 + b^2$ is a crucial formula to remember when solving for unknowns.