The values of 'm' for which a line with slope m is common tangent to the hyperbola `x^2/a^2-y^2/b^2=1` and parabola `y^2 = 4ax` can lie in interval:

Explanation

To find the values of 'm' for which a line with slope 'm' is a common tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and the parabola \( y^2 = 4ax \), we follow these steps:

### Step 1: Write the equations of the tangents
The equation of the tangent to the hyperbola is given by:
\[ y = mx \pm \sqrt{a^2m^2 - b^2} \]
The equation of the tangent to the parabola is given by:
\[ y = mx + \frac{a}{m} \]

### Step 2: Set the tangents equal
Since the line is a common tangent to both curves, we set the two equations equal to each other:
\[ mx + \frac{a}{m} = mx \pm \sqrt{a^2m^2 - b^2} \]

### Step 3: Simplify the equation
By eliminating \( mx \) from both sides, we have:
\[ \frac{a}{m} = \pm \sqrt{a^2m^2 - b^2} \]
Squaring both sides gives:
\[ \left(\frac{a}{m}\right)^2 = a^2m^2 - b^2 \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:
\[ \frac{a^2}{m^2} = a^2m^2 - b^2 \]
Multiplying through by \( m^2 \) (assuming \( m \neq 0 \)):
\[ a^2 = a^2m^4 - b^2m^2 \]
Rearranging gives:
\[ a^2m^4 - b^2m^2 - a^2 = 0 \]

### Step 5: Let \( t = m^2 \)
Let \( t = m^2 \). The equation becomes:
\[ a^2t^2 - b^2t - a^2 = 0 \]

### Step 6: Use the quadratic formula
Using the quadratic formula \( t = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = a^2 \), \( B = -b^2 \), and \( C = -a^2 \):
\[ t = \frac{b^2 \pm \sqrt{b^4 + 4a^4}}{2a^2} \]

### Step 7: Find \( m \)
Since \( t = m^2 \), we have:
\[ m^2 = \frac{b^2 \pm \sqrt{b^4 + 4a^4}}{2a^2} \]
Thus,
\[ m = \pm \sqrt{\frac{b^2 \pm \sqrt{b^4 + 4a^4}}{2a^2}} \]

### Step 8: Determine the intervals for \( m \)
To find the intervals for \( m \), we need to analyze the expression:
1. The term \( \sqrt{b^4 + 4a^4} \) is always positive.
2. The expression \( \frac{b^2 + \sqrt{b^4 + 4a^4}}{2a^2} \) is positive, leading to positive values for \( m \).
3. The expression \( \frac{b^2 - \sqrt{b^4 + 4a^4}}{2a^2} \) can be negative or zero, leading to potential negative values for \( m \).

### Final Result
The values of \( m \) lie in the intervals:
\[ (-\infty, -\sqrt{\frac{b^2 - \sqrt{b^4 + 4a^4}}{2a^2}}) \cup (\sqrt{\frac{b^2 + \sqrt{b^4 + 4a^4}}{2a^2}}, \infty) \]