f the normal at the point `P (at_1, 2at_1)` meets the parabola `y^2=4ax` aguin at `(at_2, 2at_2),` then

Options

A. `t_(1)t_(2)=-1`
B. `t_(1)t_(2)=1`
C. `t_(1)t_(2)=2`
D. `t_(2)t_(2)=-2`

Answer

To solve the problem, we need to establish the relationship between the parameters \( t_1 \) and \( t_2 \) of the points on the parabola \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Understanding the Points on the Parabola**:
The points on the parabola \( y^2 = 4ax \) can be represented in terms of the parameter \( t \). The point corresponding to parameter \( t_1 \) is given by:
\[ P(t_1) = (at_1^2, 2at_1) \]
Here, the coordinates are \( (at_1^2, 2at_1) \).

2. **Finding the Normal at Point \( P(t_1) \)**:
The equation of the normal to the parabola at the point \( P(t_1) \) is given by:
\[ y - 2at_1 = -\frac{1}{t_1}(x - at_1^2) \]
Rearranging this, we get:
\[ y = -\frac{1}{t_1}x + at_1 + 2at_1 \]
Simplifying further, the equation of the normal becomes:
\[ y = -\frac{1}{t_1}x + 3at_1 \]

3. **Finding the Intersection of the Normal with the Parabola**:
To find where this normal meets the parabola again, we substitute \( y \) from the normal equation into the parabola's equation \( y^2 = 4ax \):
\[ \left(-\frac{1}{t_1}x + 3at_1\right)^2 = 4ax \]
Expanding this gives:
\[ \frac{1}{t_1^2}x^2 - \frac{6a}{t_1}x + 9a^2t_1^2 = 4ax \]
Rearranging yields:
\[ \frac{1}{t_1^2}x^2 - \left(\frac{6a}{t_1} + 4a\right)x + 9a^2t_1^2 = 0 \]

4. **Using the Quadratic Formula**:
The above equation is a quadratic in \( x \). The roots of this equation correspond to the points where the normal intersects the parabola. Let the roots be \( x_1 = at_1^2 \) and \( x_2 = at_2^2 \). By Vieta's formulas, we know:
\[ x_1 + x_2 = \frac{6a}{t_1} + 4a \]
and
\[ x_1 x_2 = 9a^2t_1^2 \]

5. **Establishing the Relationship Between \( t_1 \) and \( t_2 \)**:
From the properties of the normal, we have the relationship:
\[ t_2 = -t_1 - \frac{2}{t_1} \]
This gives us a direct relationship between \( t_1 \) and \( t_2 \).

6. **Final Relationship**:
Setting the two expressions for \( t_2 \) equal to each other, we can derive:
\[ t_1 t_2 = 2 \]

### Conclusion:
Thus, the relationship between \( t_1 \) and \( t_2 \) is:
\[ t_1 t_2 = 2 \]