If the normal at(1, 2) on the parabola `y^(2)=4x` meets the parabola again at the point `(t^(2), 2t)` then the value of t, is

Options

A. 1
B. 3
C. -3
D. -1

Answer

To solve the problem, we need to find the value of \( t \) such that the normal at the point \( (1, 2) \) on the parabola \( y^2 = 4x \) meets the parabola again at the point \( (t^2, 2t) \).

### Step-by-Step Solution:

1. **Identify the parameters of the parabola**:
The given parabola is \( y^2 = 4x \). Here, we can identify \( a = 1 \) (since \( 4a = 4 \)).

2. **Identify the point on the parabola**:
The point \( (1, 2) \) can be expressed in terms of \( t_1 \):
\[ t_1^2 = 1 \quad \text{and} \quad 2t_1 = 2 \]
From \( 2t_1 = 2 \), we find \( t_1 = 1 \).

3. **Use the relationship of normals**:
The relationship between \( t_1 \) and \( t_2 \) for the normal is given by:
\[ t_2 = -t_1 - \frac{2}{t_1} \]
Substituting \( t_1 = 1 \):
\[ t_2 = -1 - \frac{2}{1} = -1 - 2 = -3 \]

4. **Find the coordinates of the point where the normal meets the parabola again**:
The coordinates corresponding to \( t_2 \) are:
\[ (t_2^2, 2t_2) = ((-3)^2, 2 \cdot -3) = (9, -6) \]

5. **Set the coordinates equal to \( (t^2, 2t) \)**:
We have:
\[ t^2 = 9 \quad \text{and} \quad 2t = -6 \]

6. **Solve for \( t \)**:
From \( t^2 = 9 \), we find:
\[ t = 3 \quad \text{or} \quad t = -3 \]
From \( 2t = -6 \), we find:
\[ t = -3 \]

7. **Conclusion**:
The value of \( t \) that satisfies both conditions is:
\[ t = -3 \]

### Final Answer:
The value of \( t \) is \( -3 \).