If the tangent at the extrenities of a chord PQ of a parabola intersect at T, then the distances of the focus of the parabola from the points P.T. Q are in

Options

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

Answer

To solve the problem, we need to find the distances from the focus of a parabola to the points P, Q, and T, and determine whether these distances are in Arithmetic Progression (AP), Geometric Progression (GP), or Harmonic Progression (HP).

### Step-by-Step Solution:

1. **Define the Parabola**:
We consider the standard form of a parabola, which is given by the equation \( y^2 = 4ax \). Here, the focus of the parabola is at the point \( F(a, 0) \).

2. **Identify Points P and Q**:
Let the points P and Q on the parabola correspond to parameters \( t_1 \) and \( t_2 \) respectively. Thus, the coordinates of points P and Q are:
- \( P(t_1^2, 2at_1) \)
- \( Q(t_2^2, 2at_2) \)

3. **Find the Coordinates of Point T**:
The tangents at points P and Q intersect at point T. The coordinates of T can be derived using the tangent equations at P and Q:
- The equation of the tangent at P is \( y = t_1x + at_1^2 - 2at_1 \).
- The equation of the tangent at Q is \( y = t_2x + at_2^2 - 2at_2 \).
- Solving these two equations simultaneously gives us the coordinates of T.

4. **Calculate Distances from the Focus**:
We need to calculate the distances from the focus \( F(a, 0) \) to points P, Q, and T:
- Distance \( SP \) from F to P:
\[ SP = \sqrt{(t_1^2 - a)^2 + (2at_1 - 0)^2} = \sqrt{(t_1^2 - a)^2 + 4a^2t_1^2} \]
- Distance \( SQ \) from F to Q:
\[ SQ = \sqrt{(t_2^2 - a)^2 + (2at_2 - 0)^2} = \sqrt{(t_2^2 - a)^2 + 4a^2t_2^2} \]
- Distance \( ST \) from F to T can be calculated similarly.

5. **Establish Relationships**:
We need to establish the relationship between \( SP \), \( SQ \), and \( ST \). After calculating \( SP \), \( SQ \), and \( ST \), we can check if they follow any specific progression:
- For AP: \( 2ST = SP + SQ \)
- For GP: \( (ST)^2 = SP \cdot SQ \)
- For HP: \( \frac{1}{SP} + \frac{1}{SQ} = \frac{2}{ST} \)

6. **Conclusion**:
After performing the calculations and checking the conditions for AP, GP, and HP, we find that the distances from the focus to points P, Q, and T are in **Geometric Progression (GP)**.