If `y_(1),y_(2)` are the ordinates of two points P and Q on the parabola and `y_(3)` is the ordinate of the intersection of tangents at P and Q, then

Options

A. `y_(1),y_(2),y_(3)" are in AP"`
B. `y_(1),y_(3),y_(2)" are in AP"`
C. `y_(1),y_(2),y_(3)" are in GP"`
D. `y_(1),y_(3),y_(2)" are in GP"`

Answer

To solve the problem, we need to analyze the relationship between the ordinates of points P and Q on a parabola and the ordinate of the intersection of the tangents at these points.

### Step-by-Step Solution:

1. **Identify Points on the Parabola:**
Let the points P and Q on the parabola be represented as:
- \( P(x_1, y_1) \) where \( y_1 = 2a t_1 \) (for some parameter \( t_1 \))
- \( Q(x_2, y_2) \) where \( y_2 = 2a t_2 \) (for some parameter \( t_2 \))

2. **Parameterization of Points:**
The points can be parameterized as:
- \( P(t_1) = (at_1^2, 2at_1) \)
- \( Q(t_2) = (at_2^2, 2at_2) \)

3. **Equation of Tangents:**
The equation of the tangent at point P is given by:
\[ t_1 y = x + at_1^2 \]
The equation of the tangent at point Q is given by:
\[ t_2 y = x + at_2^2 \]

4. **Finding Intersection of Tangents:**
To find the intersection of the tangents at points P and Q, we can set up the equations:
\[ t_1 y - x = at_1^2 \quad (1) \]
\[ t_2 y - x = at_2^2 \quad (2) \]

Subtracting equation (2) from equation (1):
\[ (t_1 - t_2)y = at_1^2 - at_2^2 \]

5. **Simplifying the Equation:**
This can be rearranged to:
\[ y = \frac{a(t_1^2 - t_2^2)}{t_1 - t_2} \]
Using the difference of squares:
\[ y = a(t_1 + t_2) \]

6. **Relating y to y3:**
The ordinate of the intersection point (let's denote it as \( y_3 \)) can be expressed as:
\[ y_3 = a(t_1 + t_2) \]

7. **Expressing y1 and y2:**
From our earlier definitions:
- \( y_1 = 2a t_1 \)
- \( y_2 = 2a t_2 \)

8. **Adding y1 and y2:**
Now, we can add \( y_1 \) and \( y_2 \):
\[ y_1 + y_2 = 2a t_1 + 2a t_2 = 2a(t_1 + t_2) \]

9. **Finding the Arithmetic Mean:**
Dividing both sides by 2 gives:
\[ \frac{y_1 + y_2}{2} = a(t_1 + t_2) \]

10. **Final Relationship:**
Thus, we can conclude that:
\[ y_3 = \frac{y_1 + y_2}{2} \]

This means that \( y_1, y_2, y_3 \) are in Arithmetic Progression (AP).

### Conclusion:
The correct relationship is:
\[ y_3 = \frac{y_1 + y_2}{2} \]
This indicates that \( y_1, y_2, y_3 \) are in AP.