To solve the problem, we need to find out where the tangents at points P and R on the parabola \(y^2 = 4ax\) meet, given that the ordinates of points P, Q, and R are in geometric progression.
### Step-by-Step Solution:
1. **Identify the Points on the Parabola**:
Let the coordinates of points P, Q, and R be:
- \( P(t_1) = (at_1^2, 2at_1) \)
- \( Q(t_2) = (at_2^2, 2at_2) \)
- \( R(t_3) = (at_3^2, 2at_3) \)
Since the ordinates are in geometric progression, we have:
\[
2at_1, 2at_2, 2at_3 \text{ are in GP}
\]
This implies:
\[
(2at_2)^2 = (2at_1)(2at_3) \implies t_2^2 = t_1 t_3
\]
2. **Equation of Tangents at Points P and R**:
The equation of the tangent to the parabola \(y^2 = 4ax\) at a point \((at^2, 2at)\) is given by:
\[
ty = x + at^2
\]
Therefore, the equations of the tangents at points P and R are:
- For point P:
\[
t_1y = x + at_1^2
\]
- For point R:
\[
t_3y = x + at_3^2
\]
3. **Finding the Intersection of the Tangents**:
To find the intersection of these two tangents, we can solve the equations simultaneously:
\[
t_1y - x = at_1^2 \quad \text{(1)}
\]
\[
t_3y - x = at_3^2 \quad \text{(2)}
\]
Rearranging both equations gives:
\[
x = t_1y - at_1^2 \quad \text{(3)}
\]
\[
x = t_3y - at_3^2 \quad \text{(4)}
\]
Setting equations (3) and (4) equal to each other:
\[
t_1y - at_1^2 = t_3y - at_3^2
\]
Rearranging gives:
\[
(t_1 - t_3)y = at_1^2 - at_3^2
\]
\[
y = \frac{a(t_1^2 - t_3^2)}{t_1 - t_3}
\]
Using the difference of squares:
\[
y = \frac{a(t_1 - t_3)(t_1 + t_3)}{t_1 - t_3} = a(t_1 + t_3)
\]
4. **Finding the x-coordinate**:
Substitute \(y = a(t_1 + t_3)\) back into equation (3):
\[
x = t_1(a(t_1 + t_3)) - at_1^2
\]
\[
x = at_1^2 + at_1t_3 - at_1^2 = at_1t_3
\]
5. **Conclusion**:
The intersection point of the tangents at points P and R is:
\[
\left(at_1t_3, a(t_1 + t_3)\right)
\]
Since \(t_2^2 = t_1t_3\), we can express the x-coordinate as \(at_2^2\). Thus, the intersection point lies on the line through Q, which is parallel to the y-axis.
### Final Answer:
The tangents at points P and R meet on the line through Q, which is parallel to the y-axis.
