If the area of the triangle inscribed in the parabola `y^(2)=4ax` with one vertex at the vertex of the parabola and other two vertices at the extremities of a focal chord is `5a^(2)//2` , then the length of the focal chord is

Options

A. 3a
B. 5a
C. `25a//4`
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Parabola and Focal Chord
The equation of the parabola is given as \( y^2 = 4ax \). The vertex of this parabola is at the origin (0, 0), and the focus is at the point (a, 0).

### Step 2: Identify the Coordinates of the Extremities of the Focal Chord
For a parabola \( y^2 = 4ax \), the coordinates of points on the parabola can be expressed in terms of the parameter \( t \):
- Point P (corresponding to \( t_1 \)): \( (at_1^2, 2at_1) \)
- Point Q (corresponding to \( t_2 \)): \( (at_2^2, 2at_2) \)

For a focal chord, the relationship between \( t_1 \) and \( t_2 \) is given by:
\[ t_1 t_2 = -1 \]

### Step 3: Calculate the Area of Triangle OPQ
The area of triangle OPQ can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, the base is the distance between points P and Q, and the height is the perpendicular distance from the origin O to the line PQ.

The coordinates of points P and Q are:
- \( P(at_1^2, 2at_1) \)
- \( Q(at_2^2, 2at_2) \)

The area of triangle OPQ is given as \( \frac{5a^2}{2} \).

### Step 4: Find the Length of the Focal Chord
The length of the focal chord can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(at_1^2 - at_2^2)^2 + (2at_1 - 2at_2)^2} \]

### Step 5: Substitute \( t_2 = -\frac{1}{t_1} \) into the Length Formula
Using the relationship \( t_2 = -\frac{1}{t_1} \):
\[ \text{Length} = \sqrt{(at_1^2 - a(-\frac{1}{t_1})^2)^2 + (2at_1 - 2(-\frac{1}{t_1}))^2} \]

### Step 6: Simplify the Expression
Substituting and simplifying:
1. Calculate \( at_1^2 - a\frac{1}{t_1^2} \)
2. Calculate \( 2at_1 + \frac{2a}{t_1} \)

### Step 7: Use the Area Formula to Relate \( t_1 \) and \( t_2 \)
From the area of triangle OPQ:
\[ \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{5a^2}{2} \]
Using the coordinates, we can derive an equation involving \( t_1 \).

### Step 8: Solve for \( t_1 \)
From the area equation, we can derive a quadratic equation in terms of \( t_1 \) and solve for \( t_1 \).

### Step 9: Calculate the Length of the Focal Chord
Once we have \( t_1 \), we can substitute back to find the length of the focal chord.

### Final Result
After performing all calculations, we find that the length of the focal chord is \( 5a \).

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