If `y=m x+c` is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,` having eccentricity 5, then the least positive integral value of `m` is_____

Explanation

To solve the problem, we need to find the least positive integral value of \( m \) such that the line \( y = mx + c \) is tangent to the hyperbola given by

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

with an eccentricity \( e = 5 \).

### Step 1: Use the formula for eccentricity of a hyperbola
The eccentricity \( e \) of a hyperbola is given by the formula:

\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]

Given that \( e = 5 \), we can set up the equation:

\[ 5 = \sqrt{1 + \frac{b^2}{a^2}} \]

### Step 2: Square both sides to eliminate the square root
Squaring both sides gives:

\[ 25 = 1 + \frac{b^2}{a^2} \]

### Step 3: Rearrange to find \( \frac{b^2}{a^2} \)
Subtracting 1 from both sides, we have:

\[ 24 = \frac{b^2}{a^2} \]

### Step 4: Express \( b^2 \) in terms of \( a^2 \)
From the equation \( \frac{b^2}{a^2} = 24 \), we can express \( b^2 \) as:

\[ b^2 = 24a^2 \]

### Step 5: Use the condition for tangency
For the line \( y = mx + c \) to be tangent to the hyperbola, the condition is:

\[ c^2 = a^2m^2 - b^2 \]

### Step 6: Substitute \( b^2 \) in the tangency condition
Substituting \( b^2 = 24a^2 \) into the tangency condition gives:

\[ c^2 = a^2m^2 - 24a^2 \]

### Step 7: Factor out \( a^2 \)
Factoring out \( a^2 \) from the right side:

\[ c^2 = a^2(m^2 - 24) \]

### Step 8: Ensure \( c^2 \) is non-negative
Since \( c^2 \) must be greater than or equal to 0 (as it is a square), we have:

\[ m^2 - 24 \geq 0 \]

### Step 9: Solve for \( m^2 \)
This implies:

\[ m^2 \geq 24 \]

### Step 10: Find the least positive integral value of \( m \)
Taking the square root of both sides gives:

\[ m \geq \sqrt{24} = 2\sqrt{6} \approx 4.899 \]

The least positive integer greater than or equal to \( 4.899 \) is \( 5 \).

### Final Answer
Thus, the least positive integral value of \( m \) is:

\[ \boxed{5} \]

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