If `P(at_(1)^(2), 2at_(1))" and Q(at_(2)^(2), 2at_(2))` are two points on the parabola at `y^(2)=4ax`, then that area of the triangle formed by the tangents at P and Q and the chord PQ, is

Options

A. `1/2a^(2)|t_(1)-t_(2)|^(3)`
B. `1/2a^(2)|t_(1)-t_(2)|^(2)`
C. `a^(2)|t_(1)-t_(2)|^(3)`
D. none of these

Explanation

To find the area of the triangle formed by the tangents at points P and Q on the parabola \(y^2 = 4ax\) and the chord PQ, we can follow these steps:

### Step 1: Identify the Points
The points P and Q on the parabola are given as:
- \(P(at_1^2, 2at_1)\)
- \(Q(at_2^2, 2at_2)\)

### Step 2: Write the Equation of the Tangents
The equation of the tangent to the parabola \(y^2 = 4ax\) at a point \((at^2, 2at)\) is given by:
\[ ty = x + at^2 \]
Thus, the equations of the tangents at points P and Q are:
- Tangent at P: \(t_1y = x + at_1^2\)
- Tangent at Q: \(t_2y = x + at_2^2\)

### Step 3: Find the Intersection Point of the Tangents
To find the intersection point of the tangents, we can set the equations equal to each other:
\[ t_1y - at_1^2 = x \quad \text{(1)} \]
\[ t_2y - at_2^2 = x \quad \text{(2)} \]
Setting (1) equal to (2):
\[ t_1y - at_1^2 = t_2y - at_2^2 \]
Rearranging gives:
\[ (t_1 - t_2)y = at_1^2 - at_2^2 \]
Thus, we can express \(y\) as:
\[ y = \frac{a(t_1^2 - t_2^2)}{t_1 - t_2} = a(t_1 + t_2) \]

Now substituting \(y\) back into either tangent equation to find \(x\):
\[ t_1(a(t_1 + t_2)) = x + at_1^2 \]
\[ x = at_1(t_1 + t_2) - at_1^2 = at_1t_2 \]
So the intersection point \(R\) is:
\[ R(at_1t_2, a(t_1 + t_2)) \]

### Step 4: Area of Triangle Formed by Points P, Q, and R
The area \(A\) of triangle formed by points \(P\), \(Q\), and \(R\) can be calculated using the determinant formula:
\[ A = \frac{1}{2} \left| \begin{vmatrix} at_1^2 & 2at_1 & 1 \\ at_2^2 & 2at_2 & 1 \\ at_1t_2 & a(t_1 + t_2) & 1 \end{vmatrix} \right| \]

### Step 5: Calculate the Determinant
Calculating the determinant:
\[ \begin{vmatrix} at_1^2 & 2at_1 & 1 \\ at_2^2 & 2at_2 & 1 \\ at_1t_2 & a(t_1 + t_2) & 1 \end{vmatrix} \]
Using row operations, we can simplify:
1. Subtract the second row from the first.
2. Subtract the third row from the second.

After simplification, we find:
\[ = a^2(t_1 - t_2)(t_1 + t_2) \]

### Step 6: Final Area Calculation
Thus, the area of the triangle is:
\[ A = \frac{1}{2} \cdot a^2 |t_1 - t_2| \cdot |t_1 + t_2| \]
This can be expressed as:
\[ A = \frac{1}{2} a^2 |t_1 - t_2|^2 \]

### Final Answer
The area of the triangle formed by the tangents at points P and Q and the chord PQ is:
\[ \frac{1}{2} a^2 |t_1 - t_2|^2 \]