If the distance between two parallel tangents having slope `m` drawn to the hyperbola `(x^2)/9-(y^2)/(49)=1` is 2, then the value of `2|m|` is_____

Options

A. `pm5//2`
B. `pm4//5`
C. `pm7//2`
D. none of these

Explanation

To solve the problem, we need to find the value of \(2|m|\) given that the distance between two parallel tangents to the hyperbola \(\frac{x^2}{9} - \frac{y^2}{49} = 1\) with slope \(m\) is equal to 2.

### Step-by-Step Solution:

1. **Identify the parameters of the hyperbola**:
The given hyperbola is \(\frac{x^2}{9} - \frac{y^2}{49} = 1\). Here, we can identify:
\[ a^2 = 9 \quad \Rightarrow \quad a = 3 \]
\[ b^2 = 49 \quad \Rightarrow \quad b = 7 \]

2. **Write the equation of the tangents**:
The equation of the tangents to the hyperbola with slope \(m\) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Substituting \(a^2\) and \(b^2\):
\[ y = mx \pm \sqrt{9m^2 - 49} \]

3. **Determine the distance between the two tangents**:
The distance \(d\) between the two tangents can be calculated using the formula:
\[ d = \frac{2\sqrt{9m^2 - 49}}{\sqrt{m^2 + 1}} \]
According to the problem, this distance is given to be 2:
\[ \frac{2\sqrt{9m^2 - 49}}{\sqrt{m^2 + 1}} = 2 \]

4. **Simplify the equation**:
We can simplify the equation by multiplying both sides by \(\sqrt{m^2 + 1}\) and then dividing by 2:
\[ \sqrt{9m^2 - 49} = \sqrt{m^2 + 1} \]

5. **Square both sides**:
Squaring both sides gives:
\[ 9m^2 - 49 = m^2 + 1 \]

6. **Rearranging the equation**:
Rearranging the equation leads to:
\[ 9m^2 - m^2 = 49 + 1 \]
\[ 8m^2 = 50 \]

7. **Solve for \(m^2\)**:
Dividing both sides by 8:
\[ m^2 = \frac{50}{8} = \frac{25}{4} \]

8. **Find \(m\)**:
Taking the square root gives:
\[ m = \pm \frac{5}{2} \]

9. **Calculate \(2|m|\)**:
The absolute value of \(m\) is:
\[ |m| = \frac{5}{2} \]
Therefore:
\[ 2|m| = 2 \times \frac{5}{2} = 5 \]

### Final Answer:
The value of \(2|m|\) is \(5\).