If the normals at `P(t_(1))andQ(t_(2))` on the parabola meet on the same parabola, then

Options

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

Explanation

To solve the problem, we need to analyze the conditions under which the normals at points \( P(t_1) \) and \( Q(t_2) \) on the parabola intersect at a point that also lies on the parabola.

### Step-by-step Solution:

1. **Understanding the Parabola**:
The standard equation of a parabola can be taken as \( y^2 = 4ax \). Points \( P(t_1) \) and \( Q(t_2) \) on the parabola can be represented in parametric form as:
- \( P(t_1) = (at_1^2, 2at_1) \)
- \( Q(t_2) = (at_2^2, 2at_2) \)

2. **Finding the Normals**:
The equation of the normal to the parabola at point \( P(t_1) \) is given by:
\[ y - 2at_1 = -\frac{1}{2a} (x - at_1^2) \]
Simplifying this, we get:
\[ y = -\frac{1}{2a}x + \left(2at_1 + \frac{at_1^2}{2a}\right) \]
This can be rewritten as:
\[ y = -\frac{1}{2a}x + \frac{t_1^2 + 4t_1}{2} \]

Similarly, the equation of the normal at point \( Q(t_2) \) can be derived:
\[ y = -\frac{1}{2a}x + \frac{t_2^2 + 4t_2}{2} \]

3. **Finding the Intersection Point**:
Let the normals intersect at point \( R(t_3) \) on the parabola. The coordinates of \( R \) can be expressed as:
\[ R(t_3) = (at_3^2, 2at_3) \]

4. **Setting Up the Equations**:
Since both normals intersect at the same point \( R(t_3) \), we can set the equations of the normals equal to each other:
\[ -\frac{1}{2a}x + \frac{t_1^2 + 4t_1}{2} = -\frac{1}{2a}x + \frac{t_2^2 + 4t_2}{2} \]
This simplifies to:
\[ \frac{t_1^2 + 4t_1}{2} = \frac{t_2^2 + 4t_2}{2} \]

5. **Cross Multiplying**:
Cross-multiplying gives:
\[ t_1^2 + 4t_1 = t_2^2 + 4t_2 \]

6. **Rearranging the Equation**:
Rearranging this equation leads to:
\[ t_1^2 - t_2^2 + 4(t_1 - t_2) = 0 \]
Factoring gives:
\[ (t_1 - t_2)(t_1 + t_2 + 4) = 0 \]

7. **Finding the Relationship**:
From the factored equation, we have two cases:
- \( t_1 - t_2 = 0 \) (which implies \( t_1 = t_2 \))
- \( t_1 + t_2 + 4 = 0 \) (which implies \( t_1 + t_2 = -4 \))

8. **Final Result**:
The relationship we derived indicates that if the normals intersect on the parabola, then:
\[ t_1 t_2 = 2 \]

### Conclusion:
Thus, the final answer is that if the normals at points \( P(t_1) \) and \( Q(t_2) \) on the parabola meet on the same parabola, then \( t_1 t_2 = 2 \).

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