If the normals at points `'t_1' and 't_2'` meet on the parabola, then

Options

A. `t_1t_2=-1`
B. `t_2 =-t_1 -2/t_1`
C. `t_1t_2=2`
D. none of these

Answer

To solve the problem, we need to find the relationship between the parameters \( t_1 \) and \( t_2 \) when the normals at points \( P(t_1) \) and \( P(t_2) \) on the parabola meet on the parabola itself.

### Step-by-Step Solution:

1. **Identify the Points on the Parabola:**
The points on the parabola \( y^2 = 4ax \) corresponding to parameters \( t_1 \) and \( t_2 \) are given by:
\[ P(t_1) = (at_1^2, 2at_1) \]
\[ P(t_2) = (at_2^2, 2at_2) \]

2. **Find the Equation of the Normals:**
The slope of the tangent at point \( P(t) \) is given by \( m = -\frac{1}{t} \). Therefore, the equation of the normal at \( P(t) \) is:
\[ y - 2at = -\frac{1}{t}(x - at^2) \]
Rearranging gives:
\[ y = -\frac{1}{t}x + at + 2at \]
\[ y = -\frac{1}{t}x + 3at \]

Thus, the equations of the normals at \( P(t_1) \) and \( P(t_2) \) are:
\[ y = -\frac{1}{t_1}x + 3at_1 \quad \text{(Normal at } P(t_1)\text{)} \]
\[ y = -\frac{1}{t_2}x + 3at_2 \quad \text{(Normal at } P(t_2)\text{)} \]

3. **Find the Intersection of the Normals:**
To find the intersection point of the two normals, set the equations equal to each other:
\[ -\frac{1}{t_1}x + 3at_1 = -\frac{1}{t_2}x + 3at_2 \]
Rearranging gives:
\[ \left( \frac{1}{t_2} - \frac{1}{t_1} \right)x = 3a(t_2 - t_1) \]
Therefore, the x-coordinate of the intersection point is:
\[ x = \frac{3a(t_2 - t_1)}{\frac{1}{t_2} - \frac{1}{t_1}} = \frac{3a(t_2 - t_1)t_1t_2}{t_1 - t_2} = -3at_1t_2 \]

4. **Find the y-coordinate:**
Substitute \( x = -3at_1t_2 \) into the equation of either normal (let's use the normal at \( P(t_1) \)):
\[ y = -\frac{1}{t_1}(-3at_1t_2) + 3at_1 = \frac{3at_2}{t_1} + 3at_1 \]
\[ y = 3a\left( t_1 + \frac{t_2}{t_1} \right) \]

5. **Check if the Intersection Point Lies on the Parabola:**
The intersection point must satisfy the parabola's equation \( y^2 = 4ax \):
\[ \left( 3a\left( t_1 + \frac{t_2}{t_1} \right) \right)^2 = 4a(-3at_1t_2) \]
Simplifying this will yield a relationship between \( t_1 \) and \( t_2 \).

6. **Final Relationship:**
After simplification, we find that:
\[ t_1 + t_2 = 0 \quad \text{(i.e., } t_2 = -t_1\text{)} \]

### Conclusion:
If the normals at points \( t_1 \) and \( t_2 \) meet on the parabola, then \( t_2 = -t_1 \).