To understand why the correct answer is option B, let's analyze the given information step by step.

Given:
- AB and AC are tangents to the parabola y^2 = 4ax.
- p1, p2, and p3 are the lengths of the perpendiculars from points A, B, and C respectively on any tangent to the curve.

We need to determine the relationship between p2, p1, and p3.

Analysis:

1. Equation of the Parabola:
The equation of the parabola is given by y^2 = 4ax, where "a" is a constant.

2. Tangents to the Parabola:
The tangents to the parabola y^2 = 4ax can be found using the derivative of the equation with respect to x.

Differentiating y^2 = 4ax with respect to x, we get:
2yy' = 4a
y' = 2a/y

The slope of the tangent at any point (x, y) on the parabola is given by y' = 2a/y.

Since the slope of the tangent is perpendicular to the radius of the parabola, the slope of the radius is given by -1/y'.

Therefore, the equation of the tangent at any point (x, y) on the parabola is:
y - y1 = (-1/y1)(x - x1), where (x1, y1) is a point on the parabola.

3. Perpendicular Lengths:
We are given p1, p2, and p3 as the lengths of the perpendiculars from points A, B, and C respectively on any tangent to the curve.

Let's consider a general tangent to the parabola with equation y - y1 = (-1/y1)(x - x1).

The coordinates of points A, B, and C can be written as:
A(x1, y1)
B(x2, y2)
C(x3, y3)

The lengths of the perpendiculars from points A, B, and C on this tangent are:
p1 = |y - y1|
p2 = |y2 - y1|
p3 = |y3 - y1|

To find the relationship between p2, p1, and p3, we need to eliminate y from the above equations.

4. Eliminating y:
From the equation of the tangent, we can write:
y = (-1/y1)(x - x1) + y1

Substituting this value of y in the equations for p1, p2, and p3, we get:
p1 = |-1/y1(x - x1)|
p2 = |(-1/y1)(x2 - x1)|
p3 = |(-1/y1)(x3 - x1)|

Simplifying these equations, we have:
p2 = p1(x2 - x1)/x - x1
p3 = p1(x3 - x1)/x - x1

We can see that p2/p1 = (x2 - x1)/(x - x1) and p3/p1 = (x3 - x1)/(x - x1).

Since p2, p1, and p3 are in a geometric progression (G.P.), the ratio of any two consecutive terms