If `(5, 12)" and " (24,7)` are the focii of a hyperbola passing through origin, then

Options

A. `e = sqrt(386)/13`
B. `e = sqrt(386)/12`
C. `L.R = (121)/6` units
D. length of conjugate axis is `sqrt(242)` units

Answer

To solve the problem, we need to find the eccentricity, length of the latus rectum, and the length of the conjugate axis of the hyperbola that passes through the origin and has foci at points (5, 12) and (24, 7).

### Step 1: Find the coordinates of the foci
The coordinates of the foci are given as:
- \( F_1 = (5, 12) \)
- \( F_2 = (24, 7) \)

### Step 2: Calculate the distance between the foci
The distance \( d \) between the two foci \( F_1 \) and \( F_2 \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(24 - 5)^2 + (7 - 12)^2} = \sqrt{(19)^2 + (-5)^2} = \sqrt{361 + 25} = \sqrt{386} \]

### Step 3: Determine the value of \( 2c \)
The distance between the foci is \( 2c \), so:
\[ 2c = \sqrt{386} \implies c = \frac{\sqrt{386}}{2} \]

### Step 4: Find the distance from the origin to the foci
The distance from the origin (0, 0) to \( F_1 \) and \( F_2 \) is:
\[ PF_1 = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
\[ PF_2 = \sqrt{(24 - 0)^2 + (7 - 0)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 5: Use the property of hyperbola
For a hyperbola, the property states:
\[ |PF_1 - PF_2| = 2a \]
Substituting the distances:
\[ |13 - 25| = 2a \implies 12 = 2a \implies a = 6 \]

### Step 6: Calculate the eccentricity \( e \)
The eccentricity \( e \) of a hyperbola is given by:
\[ e = \frac{c}{a} \]
Substituting the known values:
\[ c = \frac{\sqrt{386}}{2}, \quad a = 6 \]
\[ e = \frac{\frac{\sqrt{386}}{2}}{6} = \frac{\sqrt{386}}{12} \]

### Step 7: Calculate the length of the latus rectum
The length of the latus rectum \( L \) is given by:
\[ L = \frac{2b^2}{a} \]
We know \( b^2 = a^2(e^2 - 1) \). First, we calculate \( e^2 \):
\[ e^2 = \left(\frac{\sqrt{386}}{12}\right)^2 = \frac{386}{144} \]
Now, calculate \( b^2 \):
\[ b^2 = 6^2\left(\frac{386}{144} - 1\right) = 36\left(\frac{386 - 144}{144}\right) = 36\left(\frac{242}{144}\right) = \frac{8712}{144} \]
Now substituting \( b^2 \) into the latus rectum formula:
\[ L = \frac{2 \cdot \frac{8712}{144}}{6} = \frac{8712}{432} = \frac{2178}{108} = 20.25 \]

### Step 8: Calculate the length of the conjugate axis
The length of the conjugate axis \( 2b \) is given by:
\[ b = \sqrt{a^2(e^2 - 1)} = \sqrt{36 \left(\frac{386}{144} - 1\right)} = \sqrt{36 \cdot \frac{242}{144}} = \sqrt{\frac{8712}{144}} = \frac{93}{12} \]
Thus, the length of the conjugate axis is:
\[ 2b = 2 \cdot \frac{93}{12} = \frac{186}{12} = 15.5 \]

### Final Results
- Eccentricity \( e = \frac{\sqrt{386}}{12} \)
- Length of the latus rectum \( L = 20.25 \)
- Length of the conjugate axis \( = 15.5 \)