If the distance of the point `(alpha,2)` from its chord of contact w.r.t. the parabola `y^2=4x` is 4, then find the value of `alphadot`

Explanation

To solve the problem, we need to find the value of \( \alpha \) given that the distance of the point \( (\alpha, 2) \) from its chord of contact with respect to the parabola \( y^2 = 4x \) is 4.

### Step-by-step Solution:

1. **Identify the Chord of Contact Equation**:
For the parabola \( y^2 = 4x \), the equation of the chord of contact from a point \( (x_1, y_1) \) is given by:
\[ yy_1 = 2(x + x_1) \]
Here, substituting \( (x_1, y_1) = (\alpha, 2) \):
\[ y \cdot 2 = 2(x + \alpha) \implies 2y = 2x + 2\alpha \implies y - x - \alpha = 0 \]

2. **Distance from the Point to the Line**:
The distance \( d \) from the point \( (\alpha, 2) \) to the line \( y - x - \alpha = 0 \) can be calculated using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = -1 \), \( B = 1 \), and \( C = -\alpha \). Thus, substituting \( (x_1, y_1) = (\alpha, 2) \):
\[ d = \frac{|-1 \cdot \alpha + 1 \cdot 2 - \alpha|}{\sqrt{(-1)^2 + 1^2}} = \frac{|2 - 2\alpha|}{\sqrt{2}} \]

3. **Set the Distance Equal to 4**:
According to the problem, this distance is equal to 4:
\[ \frac{|2 - 2\alpha|}{\sqrt{2}} = 4 \]
Multiplying both sides by \( \sqrt{2} \):
\[ |2 - 2\alpha| = 4\sqrt{2} \]

4. **Remove the Absolute Value**:
This leads to two cases:
- Case 1: \( 2 - 2\alpha = 4\sqrt{2} \)
- Case 2: \( 2 - 2\alpha = -4\sqrt{2} \)

5. **Solve Case 1**:
For the first case:
\[ 2 - 2\alpha = 4\sqrt{2} \implies -2\alpha = 4\sqrt{2} - 2 \implies \alpha = 1 - 2\sqrt{2} \]

6. **Solve Case 2**:
For the second case:
\[ 2 - 2\alpha = -4\sqrt{2} \implies -2\alpha = -4\sqrt{2} - 2 \implies \alpha = 2 + 2\sqrt{2} \]

7. **Final Values of \( \alpha \)**:
Thus, the two possible values of \( \alpha \) are:
\[ \alpha = 1 - 2\sqrt{2} \quad \text{and} \quad \alpha = 2 + 2\sqrt{2} \]