The normal at the point `(b t_(1)^(2), 2b t_(1))` on a parabola meets the parabola again in the point `(b t_(2)^(2), 2b t_(2))` then

Options

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

Answer

To solve the problem, we need to find the relationship between \( t_1 \) and \( t_2 \) given that the normal at the point \( (bt_1^2, 2bt_1) \) on the parabola \( y^2 = 4bx \) meets the parabola again at the point \( (bt_2^2, 2bt_2) \).

### Step-by-Step Solution:

1. **Identify the Parabola**:
The equation of the parabola is given as \( y^2 = 4bx \).

2. **Find the Coordinates of the Point on the Parabola**:
The point on the parabola is given as \( (bt_1^2, 2bt_1) \).

3. **Equation of the Normal**:
The equation of the normal at a point \( (bt^2, 2bt) \) on the parabola is given by:
\[ y = -tx + 2bt + bt^3 \]
For our point \( (bt_1^2, 2bt_1) \), substituting \( t = t_1 \):
\[ y = -t_1x + 2bt_1 + bt_1^3 \]

4. **Substituting the Coordinates of the Second Point**:
The normal intersects the parabola again at the point \( (bt_2^2, 2bt_2) \). We substitute these coordinates into the normal equation:
\[ 2bt_2 = -t_1(bt_2^2) + 2bt_1 + bt_1^3 \]

5. **Simplifying the Equation**:
Dividing through by \( b \) (assuming \( b \neq 0 \)):
\[ 2t_2 = -t_1t_2^2 + 2t_1 + t_1^3 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ t_1t_2^2 + 2t_2 - t_1^3 - 2t_1 = 0 \]

7. **Using the Quadratic Formula**:
This is a quadratic equation in \( t_2 \):
\[ t_1t_2^2 + 2t_2 + (-t_1^3 - 2t_1) = 0 \]
Using the quadratic formula \( t_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = t_1 \), \( b = 2 \), and \( c = -t_1^3 - 2t_1 \).

8. **Finding the Discriminant**:
The discriminant \( D \) is given by:
\[ D = 2^2 - 4(t_1)(-t_1^3 - 2t_1) = 4 + 4t_1(t_1^3 + 2t_1) = 4 + 4t_1^4 + 8t_1^2 \]

9. **Condition for Real Roots**:
For \( t_2 \) to be real, \( D \) must be non-negative. This gives us the relation between \( t_1 \) and \( t_2 \).

10. **Final Relationship**:
After simplification, we find that:
\[ t_1 + t_2 = -\frac{2}{t_1} \]
Thus, the relationship between \( t_1 \) and \( t_2 \) is established.