A tangent line L is drawn at the point (2, -4) on the parabola `y^(2) = 8x`. If the line L is also tangent to the circle `x^(2) + y^(2) = a`, then 'a' is equal to ________.

Answer

To solve the problem, we need to find the value of 'a' such that the tangent line at the point (2, -4) on the parabola \( y^2 = 8x \) is also tangent to the circle \( x^2 + y^2 = a \).

### Step-by-Step Solution:

1. **Identify the parabola and point of tangency**:
The given parabola is \( y^2 = 8x \). We need to find the equation of the tangent line at the point \( (2, -4) \).

2. **Find the value of 'a' for the parabola**:
The standard form for the parabola \( y^2 = 4ax \) gives us \( 4a = 8 \), so \( a = 2 \).

3. **Write the equation of the tangent line**:
The equation of the tangent line to the parabola at the point \( (x_1, y_1) \) can be written as:
\[ yy_1 = 2a(x + x_1) \]
Substituting \( (x_1, y_1) = (2, -4) \) and \( a = 2 \):
\[ y(-4) = 4(2)(x + 2) \]
Simplifying gives:
\[ -4y = 8(x + 2) \]
Rearranging leads to:
\[ y = -2x - 2 \]

4. **Express the tangent line in slope-intercept form**:
The tangent line can be expressed as:
\[ y = -x - 2 \]
Here, the slope \( m = -1 \) and the y-intercept \( c = -2 \).

5. **Find the condition for tangency with the circle**:
The circle is given by \( x^2 + y^2 = a \). For the line to be tangent to the circle, we can use the condition:
\[ R^2(1 + m^2) = c^2 \]
where \( R^2 = a \), \( m = -1 \), and \( c = -2 \).

6. **Substitute values into the tangency condition**:
Plugging in the values:
\[ a(1 + (-1)^2) = (-2)^2 \]
This simplifies to:
\[ a(1 + 1) = 4 \]
Therefore:
\[ 2a = 4 \implies a = 2 \]

### Final Answer:
Thus, the value of 'a' is \( \boxed{2} \).