If `y = mx + 1` is a tangent to the hyperbola `4x^(2) - 25y^(2) = 100`, find the value of 25 `m^(4) + 5m^(2) + 1`.

Answer

To solve the problem, we need to find the value of \( 25m^4 + 5m^2 + 1 \) given that the line \( y = mx + 1 \) is a tangent to the hyperbola \( 4x^2 - 25y^2 = 100 \).

### Step 1: Rewrite the hyperbola equation
The equation of the hyperbola can be rewritten in standard form:
\[ \frac{x^2}{25} - \frac{y^2}{4} = 1 \]
This shows that \( a^2 = 25 \) and \( b^2 = 4 \), so \( a = 5 \) and \( b = 2 \).

### Step 2: Use the tangent line formula
For a hyperbola, the equation of the tangent line at a point \((x_1, y_1)\) is given by:
\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \]
If the slope of the tangent is \( m \), the equation can also be expressed as:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
In our case, the line is given as \( y = mx + 1 \). Thus, we equate:
\[ 1 = \pm \sqrt{25m^2 - 4} \]

### Step 3: Solve for \( m \)
Squaring both sides gives us:
\[ 1^2 = 25m^2 - 4 \]
This simplifies to:
\[ 1 = 25m^2 - 4 \]
\[ 25m^2 = 5 \]
\[ m^2 = \frac{1}{5} \]

### Step 4: Substitute \( m^2 \) into the expression
We need to find \( 25m^4 + 5m^2 + 1 \). First, calculate \( m^4 \):
\[ m^4 = (m^2)^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \]
Now substitute \( m^2 \) and \( m^4 \) into the expression:
\[ 25m^4 = 25 \cdot \frac{1}{25} = 1 \]
\[ 5m^2 = 5 \cdot \frac{1}{5} = 1 \]
Thus:
\[ 25m^4 + 5m^2 + 1 = 1 + 1 + 1 = 3 \]

### Final Answer
The value of \( 25m^4 + 5m^2 + 1 \) is \( \boxed{3} \).