A triangle ABC of area `Delta` is inscribed in the parabola `y^2 = 4ax` such that the vertex A lies at the vertex of the parabola and BC is a focal chord. The differences of the distances of B and C from the axis of the parabola is

Options

A. `(2Delta)/a`
B. `(2Delta)/a^2`
C. `a/(2Delta)`
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the Coordinates of Points B and C
Since triangle ABC is inscribed in the parabola \(y^2 = 4ax\), we know that point A (the vertex) is at the origin, \(A(0, 0)\). The points B and C lie on the parabola and are defined as focal chord points.

Let:
- Point B be represented by the parameter \(t_1\):
\[ B(at_1^2, 2at_1) \]
- Point C be represented by the parameter \(t_2\):
\[ C(at_2^2, 2at_2) \]

### Step 2: Use the Focal Chord Condition
For a focal chord, the relationship between the parameters \(t_1\) and \(t_2\) is given by:
\[ t_1 t_2 = -1 \]
From this, we can express \(t_2\) in terms of \(t_1\):
\[ t_2 = -\frac{1}{t_1} \]

### Step 3: Calculate the Coordinates of Points B and C
Substituting \(t_2\) into the coordinates for point C, we have:
\[ C\left(a\left(-\frac{1}{t_1}\right)^2, 2a\left(-\frac{1}{t_1}\right)\right) = C\left(\frac{a}{t_1^2}, -\frac{2a}{t_1}\right) \]

### Step 4: Determine the Distances from the Axis of the Parabola
The distance of a point from the axis of the parabola (the y-axis) is given by the x-coordinate of that point. Thus:
- The distance of point B from the axis is:
\[ d_B = at_1^2 \]
- The distance of point C from the axis is:
\[ d_C = \frac{a}{t_1^2} \]

### Step 5: Find the Difference in Distances
Now, we need to find the difference in distances of points B and C from the axis:
\[ \text{Difference} = d_B - d_C = at_1^2 - \frac{a}{t_1^2} \]

### Step 6: Simplify the Expression
Factoring out \(a\):
\[ \text{Difference} = a\left(t_1^2 - \frac{1}{t_1^2}\right) \]
Using the identity \(x^2 - \frac{1}{x^2} = \frac{(x^2 - 1)^2}{x^2}\):
\[ \text{Difference} = a\left(\frac{(t_1^2 - 1)^2}{t_1^2}\right) \]

### Conclusion
Thus, the difference of the distances of B and C from the axis of the parabola is:
\[ \text{Difference} = a\left(t_1^2 - \frac{1}{t_1^2}\right) \]