If the tangents drawn from the point `(0, 2)` to the parabola `y^2 = 4ax` are inclined at angle `(3pi)/4` , then the value of 'a' is

Answer

To solve the problem, we need to find the value of 'a' for the parabola \( y^2 = 4ax \) given that the tangents drawn from the point \( (0, 2) \) are inclined at an angle of \( \frac{3\pi}{4} \).

### Step-by-Step Solution:

1. **Understanding the Parabola**:
The equation of the parabola is given as \( y^2 = 4ax \). This is a standard form of a parabola that opens to the right.

2. **Identifying the Point**:
We have the point \( P(0, 2) \) from which the tangents are drawn to the parabola.

3. **Finding the Angle Between Tangents**:
The angle between the two tangents is given as \( \frac{3\pi}{4} \). This means the angle between the tangents with respect to the x-axis can be calculated. Since the total angle is \( \frac{3\pi}{4} \), the angle with respect to the x-axis for one of the tangents will be \( 45^\circ \) (or \( \frac{\pi}{4} \)).

4. **Finding the Slope of the Tangent**:
The slope \( m \) of the tangent that makes an angle of \( 45^\circ \) with the x-axis is given by:
\[ m = \tan(45^\circ) = 1 \]

5. **Equation of the Tangent**:
The equation of the tangent to the parabola \( y^2 = 4ax \) in slope form is:
\[ y = mx + \frac{a}{m} \]
Substituting \( m = 1 \):
\[ y = x + a \]

6. **Substituting the Point**:
Since the tangent passes through the point \( (0, 2) \), we substitute \( x = 0 \) and \( y = 2 \) into the tangent equation:
\[ 2 = 0 + a \]
This simplifies to:
\[ a = 2 \]

### Conclusion:
The value of \( a \) is \( 2 \).