To solve the problem step by step, we will find the area of triangle \( \Delta PLL' \) where \( L = (2, 4) \) and \( L' = (2, -4) \) are the ends of the latus rectum of the parabola, and \( P \) is a point on the directrix.
### Step 1: Identify the points and the latus rectum
The points given are:
- \( L(2, 4) \)
- \( L'(2, -4) \)
### Step 2: Find the focus of the parabola
The focus \( F \) of the parabola can be found as the midpoint of points \( L \) and \( L' \).
Using the midpoint formula:
\[
F = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 2}{2}, \frac{4 + (-4)}{2} \right) = (2, 0)
\]
### Step 3: Calculate the length of the latus rectum
The length of the latus rectum can be calculated using the distance formula:
\[
\text{Length of } LL' = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 2)^2 + (-4 - 4)^2} = \sqrt{0 + 64} = 8
\]
### Step 4: Determine the value of \( A \)
The length of the latus rectum is given by the formula \( 4A \). Therefore, we can set up the equation:
\[
8 = 4A \implies A = 2
\]
### Step 5: Find the vertex of the parabola
The vertex \( V \) lies on the axis of the parabola, which is vertically aligned with the focus. The vertex is located \( A \) units to the left of the focus:
\[
V = (2 - A, 0) = (2 - 2, 0) = (0, 0)
\]
### Step 6: Find the point \( P \) on the directrix
The directrix is located \( A \) units to the left of the vertex. Thus, the coordinates of point \( P \) are:
\[
P = (0 - 2, 0) = (-2, 0)
\]
### Step 7: Calculate the area of triangle \( \Delta PLL' \)
The area of triangle \( \Delta PLL' \) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is the length of the latus rectum (8) and the height is the distance from point \( P \) to the line through points \( L \) and \( L' \) (which is the y-coordinate of \( L \) or \( L' \), which is 4).
Thus, the area is:
\[
\text{Area} = \frac{1}{2} \times 8 \times 4 = 16 \text{ square units}
\]
### Final Answer
The area of triangle \( \Delta PLL' \) is \( 16 \) square units.
---
