If A line with slope m touches the hyperbola `(x^(2))/(25)-(y^(2))/(4)` =1 and the parablola `y^(2)=20x` then the value of `25m^(4)-4m^(2)` is equal to

Options

A. 29
B. 21
C. 25
D. 4

Answer

To solve the problem, we need to find the value of \( 25m^4 - 4m^2 \) given that a line with slope \( m \) touches both the hyperbola \( \frac{x^2}{25} - \frac{y^2}{4} = 1 \) and the parabola \( y^2 = 20x \).

### Step-by-Step Solution:

1. **Identify the equations of the hyperbola and the parabola:**
- The hyperbola is given by \( \frac{x^2}{25} - \frac{y^2}{4} = 1 \).
- The parabola is given by \( y^2 = 20x \).

2. **Find the parameters of the hyperbola:**
- From the hyperbola equation, we have \( a^2 = 25 \) and \( b^2 = 4 \).
- Thus, \( a = 5 \) and \( b = 2 \).

3. **Write the equation of the tangent to the hyperbola:**
- The equation of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
- Substituting \( a^2 \) and \( b^2 \):
\[ y = mx \pm \sqrt{25m^2 - 4} \]

4. **Find the parameters of the parabola:**
- The parabola \( y^2 = 20x \) can be rewritten in the standard form \( y^2 = 4ax \) where \( 4a = 20 \).
- Thus, \( a = 5 \).

5. **Write the equation of the tangent to the parabola:**
- The equation of the tangent to the parabola \( y^2 = 4ax \) is given by:
\[ y = mx + \frac{a}{m} \]
- Substituting \( a = 5 \):
\[ y = mx + \frac{5}{m} \]

6. **Set the two tangent equations equal:**
- Since both tangents touch the same line, we equate the two expressions for \( y \):
\[ mx \pm \sqrt{25m^2 - 4} = mx + \frac{5}{m} \]
- This leads to:
\[ \pm \sqrt{25m^2 - 4} = \frac{5}{m} \]

7. **Square both sides to eliminate the square root:**
- Squaring both sides gives:
\[ 25m^2 - 4 = \left(\frac{5}{m}\right)^2 \]
- This simplifies to:
\[ 25m^2 - 4 = \frac{25}{m^2} \]

8. **Multiply through by \( m^2 \) to eliminate the fraction:**
- We multiply both sides by \( m^2 \):
\[ 25m^4 - 4m^2 = 25 \]

9. **Rearranging gives the desired expression:**
- Therefore, we have:
\[ 25m^4 - 4m^2 = 25 \]

### Final Result:
The value of \( 25m^4 - 4m^2 \) is equal to \( 25 \).