The ratio of the areas of a triangle formed with vertices `A(at_(1)^(2),2at_(1)),B(at_(2)^(2),2at_(2)),C(at_(3)^(2),2at_(3))` lies on the parabola `y^(2)=4ax` and triangle formed by the tangents at A,B,C is

Options

A. `1:2`
B. `2:1`
C. `2:3`
D. `3:2`

Answer

To find the ratio of the areas of the triangle formed by the vertices \( A(at_1^2, 2at_1) \), \( B(at_2^2, 2at_2) \), and \( C(at_3^2, 2at_3) \) to the area of the triangle formed by the tangents at points \( A, B, C \) on the parabola \( y^2 = 4ax \), we will follow these steps:

### Step 1: Find the coordinates of points A, B, and C
The coordinates of the points are given as:
- \( A(at_1^2, 2at_1) \)
- \( B(at_2^2, 2at_2) \)
- \( C(at_3^2, 2at_3) \)

### Step 2: Calculate the area of triangle ABC
The area of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points A, B, and C:
\[ \text{Area}_{ABC} = \frac{1}{2} \left| at_1^2(2at_2 - 2at_3) + at_2^2(2at_3 - 2at_1) + at_3^2(2at_1 - 2at_2) \right| \]
This simplifies to:
\[ = a \left| t_1^2(t_2 - t_3) + t_2^2(t_3 - t_1) + t_3^2(t_1 - t_2) \right| \]

### Step 3: Find the equations of the tangents at points A, B, and C
The equation of the tangent to the parabola \( y^2 = 4ax \) at point \( (at^2, 2at) \) is given by:
\[ yt = x + at^2 \]
Thus, the equations of the tangents at points A, B, and C are:
- For A: \( y t_1 = x + at_1^2 \)
- For B: \( y t_2 = x + at_2^2 \)
- For C: \( y t_3 = x + at_3^2 \)

### Step 4: Find the intersection points of the tangents
To find the intersection points \( P, Q, R \):
- Intersection of tangents at A and B:
\[ y t_1 = x + at_1^2 \quad (1) \]
\[ y t_2 = x + at_2^2 \quad (2) \]
Solving these two equations will give the coordinates of point \( P \).

- Intersection of tangents at B and C:
\[ y t_2 = x + at_2^2 \quad (2) \]
\[ y t_3 = x + at_3^2 \quad (3) \]
Solving these will give the coordinates of point \( Q \).

- Intersection of tangents at C and A:
\[ y t_3 = x + at_3^2 \quad (3) \]
\[ y t_1 = x + at_1^2 \quad (1) \]
Solving these will give the coordinates of point \( R \).

### Step 5: Calculate the area of triangle PQR
Using the same area formula as before, substitute the coordinates of points \( P, Q, R \) obtained from the intersection of the tangents.

### Step 6: Find the ratio of the areas
Finally, the ratio of the areas of the triangles ABC and PQR can be expressed as:
\[ \text{Ratio} = \frac{\text{Area}_{ABC}}{\text{Area}_{PQR}} \]