If `(5,12)a n d(24 ,7)` are the foci of a hyperbola passing through the origin, then `e=(sqrt(386))/(12)` (b) `e=(sqrt(386))/(13)` `L R=(121)/6` (d) `L R=(121)/3`

Options

A. `e=sqrt(386)/12`
B. `e=sqrt(386)/13`
C. `LR=121//6`
D. `LR=121//3`

Explanation

To solve the problem step by step, we need to find the eccentricity \( e \) of the hyperbola given its foci and that it passes through the origin.

### Step 1: Identify the coordinates of the foci
The foci of the hyperbola are given as \( S(5, 12) \) and \( S'(24, 7) \).

### Step 2: Calculate the distance between the foci
Using the distance formula, the distance \( SS' \) between the foci can be calculated as follows:

\[ SS' = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the foci:

\[ SS' = \sqrt{(24 - 5)^2 + (7 - 12)^2} = \sqrt{(19)^2 + (-5)^2} = \sqrt{361 + 25} = \sqrt{386} \]

### Step 3: Calculate the distances from the origin to each focus
Next, we calculate the distances from the origin \( P(0, 0) \) to each focus:

1. Distance \( SP \):
\[ SP = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

2. Distance \( S'P \):
\[ S'P = \sqrt{(24 - 0)^2 + (7 - 0)^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 4: Use the properties of the hyperbola
For a hyperbola, the following relationship holds:

\[ S'P - SP = 2a \]

Substituting the values we calculated:

\[ 25 - 13 = 2a \implies 12 = 2a \implies a = 6 \]

The distance between the foci \( SS' \) is also related to the eccentricity \( e \) by the formula:

\[ SS' = 2ae \]

Substituting the value of \( SS' \):

\[ \sqrt{386} = 2 \cdot 6 \cdot e \implies \sqrt{386} = 12e \]

### Step 5: Solve for eccentricity \( e \)
To find \( e \):

\[ e = \frac{\sqrt{386}}{12} \]

### Conclusion
Thus, the eccentricity \( e \) of the hyperbola is:

\[ e = \frac{\sqrt{386}}{12} \]

### Final Answer
The correct option is (a) \( e = \frac{\sqrt{386}}{12} \).