find the equation of hyperabola where foci are (0,12) and (0,-12)and the length of the latus rectum is 36

Explanation

To find the equation of the hyperbola with given foci and length of the latus rectum, we can follow these steps:

### Step 1: Identify the center and orientation of the hyperbola
The foci of the hyperbola are given as (0, 12) and (0, -12). This indicates that the hyperbola is vertically oriented and centered at the origin (0, 0).

### Step 2: Determine the distance to the foci (c)
The distance from the center to each focus is denoted as \( c \). Since the foci are at (0, 12) and (0, -12), we have:
\[ c = 12 \]

### Step 3: Use the length of the latus rectum to find \( a \)
The length of the latus rectum (L) for a hyperbola is given by the formula:
\[ L = \frac{2b^2}{a} \]
We are given that the length of the latus rectum is 36. Therefore, we can set up the equation:
\[ 36 = \frac{2b^2}{a} \]
This simplifies to:
\[ b^2 = 18a \]

### Step 4: Relate \( a \), \( b \), and \( c \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c^2 = a^2 + b^2 \]
Substituting \( c = 12 \):
\[ 12^2 = a^2 + b^2 \]
This simplifies to:
\[ 144 = a^2 + b^2 \]

### Step 5: Substitute \( b^2 \) in terms of \( a \)
From Step 3, we have \( b^2 = 18a \). Substitute this into the equation from Step 4:
\[ 144 = a^2 + 18a \]

### Step 6: Rearrange and solve for \( a \)
Rearranging gives us:
\[ a^2 + 18a - 144 = 0 \]
Now we can solve this quadratic equation using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 18 \), and \( c = -144 \):
\[ a = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
\[ a = \frac{-18 \pm \sqrt{324 + 576}}{2} \]
\[ a = \frac{-18 \pm \sqrt{900}}{2} \]
\[ a = \frac{-18 \pm 30}{2} \]

Calculating the two possible values for \( a \):
1. \( a = \frac{12}{2} = 6 \)
2. \( a = \frac{-48}{2} = -24 \) (not valid since \( a \) must be positive)

Thus, we have:
\[ a = 6 \]

### Step 7: Find \( b^2 \)
Now substitute \( a \) back into the equation for \( b^2 \):
\[ b^2 = 18a = 18 \cdot 6 = 108 \]

### Step 8: Write the equation of the hyperbola
The standard form of the equation of a vertically oriented hyperbola is:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]
Substituting \( a^2 = 36 \) and \( b^2 = 108 \):
\[ \frac{y^2}{36} - \frac{x^2}{108} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{y^2}{36} - \frac{x^2}{108} = 1 \]

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