The area of triangle formed by tangents at the parametrie points `t_(1),t_(2)` and `t_(3)`, on `y^(2) = 4ax` is k`|(t_(1)-t_(2)) (t_(2)-t_(1)) (t_(3)-t_(1))|` then K =

Options

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

Answer

To solve the problem of finding the area of the triangle formed by the tangents at the parametric points \( t_1, t_2, \) and \( t_3 \) on the parabola \( y^2 = 4ax \), we can follow these steps:

### Step 1: Identify the points on the parabola
The points on the parabola corresponding to the parameters \( t_1, t_2, \) and \( t_3 \) are given by:
- \( P_1(t_1) = (at_1^2, 2at_1) \)
- \( P_2(t_2) = (at_2^2, 2at_2) \)
- \( P_3(t_3) = (at_3^2, 2at_3) \)

### Step 2: Write the equations of the tangents
The equation of the tangent to the parabola at a point \( (at^2, 2at) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
For the points \( P_1, P_2, \) and \( P_3 \), the equations of the tangents are:
- For \( P_1(t_1) \): \( yy_1 = 2a(x + at_1^2) \)
- For \( P_2(t_2) \): \( yy_2 = 2a(x + at_2^2) \)
- For \( P_3(t_3) \): \( yy_3 = 2a(x + at_3^2) \)

### Step 3: Set up the area formula
The area \( A \) of the triangle formed by the tangents can be calculated using the determinant formula:
\[ A = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right| \]
Substituting the coordinates of the points:
\[ A = \frac{1}{2} \left| \begin{array}{ccc} at_1^2 & 2at_1 & 1 \\ at_2^2 & 2at_2 & 1 \\ at_3^2 & 2at_3 & 1 \end{array} \right| \]

### Step 4: Calculate the determinant
Calculating the determinant, we have:
\[ A = \frac{1}{2} \left| a \begin{array}{ccc} t_1^2 & 2t_1 & 1 \\ t_2^2 & 2t_2 & 1 \\ t_3^2 & 2t_3 & 1 \end{array} \right| \]
Factoring out \( a \):
\[ A = \frac{a}{2} \left| \begin{array}{ccc} t_1^2 & 2t_1 & 1 \\ t_2^2 & 2t_2 & 1 \\ t_3^2 & 2t_3 & 1 \end{array} \right| \]

### Step 5: Evaluate the determinant
Using the properties of determinants, we can simplify:
\[ \left| \begin{array}{ccc} t_1^2 & 2t_1 & 1 \\ t_2^2 & 2t_2 & 1 \\ t_3^2 & 2t_3 & 1 \end{array} \right| = 2(t_1 - t_2)(t_2 - t_3)(t_3 - t_1) \]
Thus, we can express the area as:
\[ A = \frac{a}{2} \cdot 2(t_1 - t_2)(t_2 - t_3)(t_3 - t_1) = a(t_1 - t_2)(t_2 - t_3)(t_3 - t_1) \]

### Step 6: Relate area to the given expression
The problem states that the area can be expressed as:
\[ A = k |(t_1 - t_2)(t_2 - t_3)(t_3 - t_1)| \]
From our calculations, we have:
\[ A = a |(t_1 - t_2)(t_2 - t_3)(t_3 - t_1)| \]
Thus, we can equate:
\[ k = a \]

### Conclusion
The value of \( k \) is \( a \).