Tangents at point B and C on the parabola `y^2=4ax` intersect at A. The perpendiculars from points A, B and C to any other tangent of the parabola are in:

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To solve the problem, we need to analyze the tangents at points B and C on the parabola \( y^2 = 4ax \) and determine the relationship between the perpendicular distances from points A, B, and C to any other tangent of the parabola.

### Step 1: Identify Points B and C
Let the points B and C on the parabola be represented as:
- \( B(t_1) = (at_1^2, 2at_1) \)
- \( C(t_2) = (at_2^2, 2at_2) \)

### Step 2: Find the Intersection Point A
The tangents at points B and C intersect at point A. The coordinates of point A can be found using the formula for the intersection of tangents:
\[ A = (at_1t_2, a(t_1 + t_2)) \]

### Step 3: Write the Equation of the Tangent
The general equation of a tangent to the parabola \( y^2 = 4ax \) can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

### Step 4: Calculate Perpendicular Distances
Now we need to calculate the perpendicular distances from points A, B, and C to the tangent line \( y = mx + \frac{a}{m} \).

1. **Perpendicular distance from A to the tangent:**
\[ p_1 = \frac{|y_A - mx_A - \frac{a}{m}|}{\sqrt{1 + m^2}} = \frac{|a(t_1 + t_2) - m(at_1t_2) - \frac{a}{m}|}{\sqrt{1 + m^2}} \]

2. **Perpendicular distance from B to the tangent:**
\[ p_2 = \frac{|y_B - mx_B - \frac{a}{m}|}{\sqrt{1 + m^2}} = \frac{|2at_1 - m(at_1^2) - \frac{a}{m}|}{\sqrt{1 + m^2}} \]

3. **Perpendicular distance from C to the tangent:**
\[ p_3 = \frac{|y_C - mx_C - \frac{a}{m}|}{\sqrt{1 + m^2}} = \frac{|2at_2 - m(at_2^2) - \frac{a}{m}|}{\sqrt{1 + m^2}} \]

### Step 5: Simplify the Distances
We can factor out \( \frac{a}{m} \) from each expression:
- For \( p_1 \):
\[ p_1 = \frac{a}{m\sqrt{1 + m^2}} \left| m(t_1 + t_2) - t_1t_2 - 1 \right| \]

- For \( p_2 \):
\[ p_2 = \frac{a}{m\sqrt{1 + m^2}} \left| m(t_1) - 1 \right|^2 \]

- For \( p_3 \):
\[ p_3 = \frac{a}{m\sqrt{1 + m^2}} \left| m(t_2) - 1 \right|^2 \]

### Step 6: Establish the Relationship
To find the relationship between \( p_1, p_2, \) and \( p_3 \), we can multiply \( p_2 \) and \( p_3 \):
\[ p_2 \cdot p_3 = \frac{a^2}{m^2(1 + m^2)} \left| m(t_1) - 1 \right|^2 \left| m(t_2) - 1 \right|^2 \]

Now, if we take the square root of \( p_1 \):
\[ \sqrt{p_1} = \frac{a}{m\sqrt{1 + m^2}} \left| \sqrt{(m(t_1 + t_2) - t_1t_2 - 1)} \right| \]

### Conclusion
Since \( p_2 \cdot p_3 = p_1 \), we conclude that \( p_1, p_2, \) and \( p_3 \) are in geometric progression (GP).

### Final Answer
The perpendiculars from points A, B, and C to any other tangent of the parabola are in **geometric progression (GP)**.

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