If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem, we need to show that the distances \( SP, ST, \) and \( SQ \) are in geometric progression (GP) when the tangents at points \( P \) and \( Q \) on the parabola \( y^2 = 4ax \) meet at point \( T \).

### Step-by-Step Solution:

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

2. **Find the Coordinates of Point T**:
The coordinates of point \( T \), where the tangents at \( P \) and \( Q \) meet, can be derived from the intersection of the tangents. The coordinates of \( T \) are:
\[ T(a, t_1 t_2, a(t_1 + t_2)) \]

3. **Calculate Distances**:
We need to find the distances \( SP, ST, \) and \( SQ \):
- **Distance \( SP \)**:
\[ SP^2 = (x_P - x_S)^2 + (y_P - y_S)^2 = (a - a)^2 + (2at_1 - 0)^2 = (2at_1)^2 = 4a^2t_1^2 \]
Thus, \( SP = 2a t_1 \).

- **Distance \( SQ \)**:
\[ SQ^2 = (x_Q - x_S)^2 + (y_Q - y_S)^2 = (a - a)^2 + (2at_2 - 0)^2 = (2at_2)^2 = 4a^2t_2^2 \]
Thus, \( SQ = 2a t_2 \).

- **Distance \( ST \)**:
\[ ST^2 = (x_T - x_S)^2 + (y_T - y_S)^2 = (at_1t_2 - 0)^2 + (a(t_1 + t_2) - 0)^2 \]
Expanding this gives:
\[ ST^2 = (at_1t_2)^2 + (a(t_1 + t_2))^2 = a^2(t_1^2 t_2^2 + (t_1 + t_2)^2) \]
Simplifying:
\[ ST^2 = a^2(t_1^2 t_2^2 + t_1^2 + 2t_1t_2 + t_2^2) = a^2((t_1^2 + 1)(t_2^2 + 1)) \]
Thus, \( ST = a \sqrt{(t_1^2 + 1)(t_2^2 + 1)} \).

4. **Establish the GP Condition**:
We need to show that \( SP, ST, SQ \) are in GP, which means:
\[ ST^2 = SP \cdot SQ \]
Substituting the values:
\[ a^2((t_1^2 + 1)(t_2^2 + 1)) = (2at_1)(2at_2) \]
Simplifying gives:
\[ a^2(t_1^2 + 1)(t_2^2 + 1) = 4a^2t_1t_2 \]
Dividing both sides by \( a^2 \) (assuming \( a \neq 0 \)):
\[ (t_1^2 + 1)(t_2^2 + 1) = 4t_1t_2 \]
This confirms that \( SP, ST, SQ \) are in GP.

### Conclusion:
Thus, we have shown that the distances \( SP, ST, SQ \) are in geometric progression.