Area of the triangle formed by the threepoints `'t_1'. 't_2' and 't_3'` on `y^2=4ax` is `K|(t_1-t_2) (t_2-t_3)(t_3-t_1)|` then `K=`

Options

A. a
B. a^(2)
C. `a^(2)/2
D. 1/4a^(2)

Explanation

To find the value of \( K \) in the area of the triangle formed by the points \( t_1, t_2, \) and \( t_3 \) on the parabola \( y^2 = 4ax \), we follow these steps:

### Step 1: Identify the coordinates of the points on the parabola
The parametric coordinates of points on the parabola \( y^2 = 4ax \) are given by:
\[ (t^2, 2t) \]
Thus, for the points \( t_1, t_2, \) and \( t_3 \), the coordinates are:
- Point \( A(t_1) = (at_1^2, 2at_1) \)
- Point \( B(t_2) = (at_2^2, 2at_2) \)
- Point \( C(t_3) = (at_3^2, 2at_3) \)

### Step 2: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by three points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 3: Substitute the coordinates into the area formula
Substituting the coordinates of points \( A, B, \) and \( C \):
\[ A = \frac{1}{2} \left| at_1^2(2at_2 - 2at_3) + at_2^2(2at_3 - 2at_1) + at_3^2(2at_1 - 2at_2) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| 2a \left( at_1^2(t_2 - t_3) + at_2^2(t_3 - t_1) + at_3^2(t_1 - t_2) \right) \right| \]
\[ = a \left| at_1^2(t_2 - t_3) + at_2^2(t_3 - t_1) + at_3^2(t_1 - t_2) \right| \]

### Step 4: Factor out common terms
Factoring out \( a \):
\[ A = a \left| t_1^2(t_2 - t_3) + t_2^2(t_3 - t_1) + t_3^2(t_1 - t_2) \right| \]

### Step 5: Use the identity for the area of the triangle
Using the identity for the area of the triangle formed by points \( t_1, t_2, t_3 \):
\[ A = K |(t_1 - t_2)(t_2 - t_3)(t_3 - t_1)| \]
We can equate the two expressions for the area:
\[ K |(t_1 - t_2)(t_2 - t_3)(t_3 - t_1)| = a \left| t_1^2(t_2 - t_3) + t_2^2(t_3 - t_1) + t_3^2(t_1 - t_2) \right| \]

### Step 6: Determine the value of \( K \)
From the analysis, we find that:
\[ K = a \]
Thus, the value of \( K \) is:
\[ \boxed{a} \]