find the standard of this hyperbola; Vertices(0,2)(0,-2) and Foci (0,4)(0,-4)

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.

Learn how to find the standard form of a hyperbola given its vertices at (0, 2) and (0, -2) and its foci at (0, 4) and (0, -4). This step-by-step solution covers key concepts in conic sections, the relationship between a, b, and c, and writing the equation of a vertical hyperbola. Suitable for students in Grades 10-12.

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