Find the equation of the hyperbola whose foci are `(0,+-4)` and latus rectum is 12.

Answer

To find the equation of the hyperbola with foci at (0, ±4) and a latus rectum of 12, we can follow these steps:

### Step 1: Identify the center and orientation of the hyperbola
The foci are given as (0, ±4), which indicates that the hyperbola opens vertically. The center of the hyperbola is at the origin (0, 0).

### Step 2: Determine the distance to the foci
The distance from the center to each focus (c) is given by the coordinates of the foci. Here, c = 4.

### Step 3: Relate the latus rectum to a and b
The length of the latus rectum (LR) for a hyperbola is given by the formula:
\[ LR = \frac{2a^2}{b} \]
We know that the latus rectum is 12, so we can set up the equation:
\[ \frac{2a^2}{b} = 12 \]
From this, we can express \( a^2 \) in terms of \( b \):
\[ 2a^2 = 12b \]
\[ a^2 = 6b \] (Equation 1)

### Step 4: Use the relationship between a, b, and c
For hyperbolas, the relationship between a, b, and c is given by:
\[ c^2 = a^2 + b^2 \]
Substituting \( c = 4 \):
\[ 4^2 = a^2 + b^2 \]
\[ 16 = a^2 + b^2 \] (Equation 2)

### Step 5: Substitute Equation 1 into Equation 2
Now, substitute \( a^2 = 6b \) from Equation 1 into Equation 2:
\[ 16 = 6b + b^2 \]
Rearranging gives us:
\[ b^2 + 6b - 16 = 0 \]

### Step 6: Solve the quadratic equation
To solve the quadratic equation \( b^2 + 6b - 16 = 0 \), we can factor it:
\[ (b + 8)(b - 2) = 0 \]
This gives us two possible solutions for b:
1. \( b = -8 \) (not valid since b must be positive)
2. \( b = 2 \)

### Step 7: Find a using b
Now that we have \( b = 2 \), we can substitute back into Equation 1 to find \( a^2 \):
\[ a^2 = 6b = 6 \times 2 = 12 \]

### Step 8: Write the equation of the hyperbola
The standard form of the equation of a hyperbola that opens vertically is:
\[ \frac{x^2}{b^2} - \frac{y^2}{a^2} = -1 \]
Substituting \( a^2 = 12 \) and \( b^2 = 4 \):
\[ \frac{x^2}{4} - \frac{y^2}{12} = -1 \]

### Step 9: Rearranging the equation
To express it in a more standard form:
\[ \frac{y^2}{12} - \frac{x^2}{4} = 1 \]
This is the final equation of the hyperbola.

### Final Answer
The equation of the hyperbola is:
\[ \frac{y^2}{12} - \frac{x^2}{4} = 1 \]

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