Let A, B, C be three points on the parabola `y^(2)=4ax` such that tangents at these points taken in pairs form a triangle PQR. Then, area `(DeltaABC)` : `(Delta PQR)`=

Options

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

Explanation

To solve the problem, we need to find the ratio of the areas of triangle ABC, formed by points A, B, and C on the parabola \(y^2 = 4ax\), and triangle PQR, formed by the tangents at these points.

### Step-by-Step Solution:

1. **Identify Points on the Parabola**:
The general points on the parabola \(y^2 = 4ax\) can be represented as:
\[ A(T_1) = (AT_1^2, 2AT_1), \quad B(T_2) = (AT_2^2, 2AT_2), \quad C(T_3) = (AT_3^2, 2AT_3) \]

2. **Find the Area of Triangle ABC**:
The area of triangle formed by points A, B, and C can be calculated using the determinant formula:
\[ \Delta ABC = \frac{1}{2} \left| \begin{array}{ccc} AT_1^2 & 2AT_1 & 1 \\ AT_2^2 & 2AT_2 & 1 \\ AT_3^2 & 2AT_3 & 1 \end{array} \right| \]
Expanding this determinant, we factor out \(A^2\):
\[ \Delta ABC = \frac{A^2}{2} \left( T_1(T_2 - T_3) + T_2(T_3 - T_1) + T_3(T_1 - T_2) \right) \]
Simplifying gives:
\[ \Delta ABC = \frac{A^2}{2} (T_1 - T_2)(T_2 - T_3)(T_3 - T_1) \]

3. **Find the Area of Triangle PQR**:
The tangents at points A, B, and C can be represented as:
\[ P = (AT_1T_2, A(T_1 + T_2)), \quad Q = (AT_2T_3, A(T_2 + T_3)), \quad R = (AT_3T_1, A(T_3 + T_1)) \]
The area of triangle PQR is given by:
\[ \Delta PQR = \frac{1}{2} \left| \begin{array}{ccc} AT_1T_2 & A(T_1 + T_2) & 1 \\ AT_2T_3 & A(T_2 + T_3) & 1 \\ AT_3T_1 & A(T_3 + T_1) & 1 \end{array} \right| \]
Similar to before, we can factor out \(A^2\):
\[ \Delta PQR = \frac{A^2}{2} (T_1 - T_2)(T_2 - T_3)(T_3 - T_1) \]

4. **Calculate the Ratio of Areas**:
Now we can find the ratio of the areas:
\[ \frac{\Delta ABC}{\Delta PQR} = \frac{\frac{A^2}{2} (T_1 - T_2)(T_2 - T_3)(T_3 - T_1)}{\frac{A^2}{2} (T_1 - T_2)(T_2 - T_3)(T_3 - T_1)} = 2 \]

Thus, the ratio of the areas is:
\[ \frac{\Delta ABC}{\Delta PQR} = 2:1 \]

### Final Answer:
The ratio of the areas \( \Delta ABC : \Delta PQR = 2 : 1 \)