To solve the problem step by step, we need to find the area of triangle OPQ formed by the tangents drawn from the origin to the parabola.
### Step 1: Determine the vertex and focus of the parabola
The vertex of the parabola is given to be at a distance of 2 units from the origin, which we can place at the point \( V(2, 0) \). The focus is at a distance of 4 units from the origin, which we can place at the point \( F(4, 0) \).
### Step 2: Write the equation of the parabola
The standard form of a parabola that opens to the right is given by:
\[
y^2 = 4a(x - h)
\]
where \( (h, k) \) is the vertex and \( a \) is the distance from the vertex to the focus. Here, \( h = 2 \), \( k = 0 \), and \( a = 2 \) (since the distance from the vertex at \( (2, 0) \) to the focus at \( (4, 0) \) is 2 units).
Thus, the equation of the parabola becomes:
\[
y^2 = 4 \cdot 2 (x - 2) \implies y^2 = 8(x - 2)
\]
### Step 3: Find the directrix of the parabola
The directrix of a parabola is given by the equation:
\[
x = h - a
\]
Substituting \( h = 2 \) and \( a = 2 \):
\[
x = 2 - 2 = 0
\]
So, the directrix is the line \( x = 0 \) (the y-axis).
### Step 4: Find the points of tangency (P and Q)
The tangents drawn from the origin (0, 0) to the parabola will meet the parabola at points P and Q. The coordinates of points P and Q can be found using the equation of the parabola.
Using the equation \( y^2 = 8(x - 2) \), we can find the points of tangency. The points of tangency can be derived from the condition for tangents from a point to a parabola. The equation for the tangents from the point \( (0, 0) \) to the parabola \( y^2 = 8(x - 2) \) can be derived as:
\[
y = mx \quad \text{(where m is the slope)}
\]
Substituting \( y = mx \) into the parabola's equation:
\[
(mx)^2 = 8(x - 2)
\]
\[
m^2 x^2 = 8x - 16
\]
Rearranging gives:
\[
m^2 x^2 - 8x + 16 = 0
\]
The discriminant of this quadratic must be zero for the tangents to touch the parabola:
\[
D = b^2 - 4ac = (-8)^2 - 4(m^2)(16) = 64 - 64m^2 = 0
\]
Thus, \( m^2 = 1 \) implies \( m = 1 \) or \( m = -1 \).
### Step 5: Find coordinates of points P and Q
For \( m = 1 \):
\[
y = x \implies y^2 = 8(x - 2) \implies x^2 = 8(x - 2)
\]
Solving gives:
\[
x^2 - 8x + 16 = 0 \implies (x - 4)^2 = 0 \implies x = 4
\]
So, \( y = 4 \) giving point \( P(4, 4) \).
For \( m = -1 \):
\[
y = -x \implies y^2 = 8(x - 2) \implies x^2 = 8(x - 2)
\]
This gives the same x-coordinate:
\[
x = 4 \implies y = -4 \text{ giving point } Q(4, -4).
\]
### Step 6: Calculate the area of triangle OPQ
The coordinates of points are:
- \( O(0, 0) \)
- \( P(4, 4) \)
- \( Q(4, -4) \)
The area \( A \) of triangle OPQ can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \( PQ \) is the vertical distance between \( P \) and \( Q \), which is:
\[
|4 - (-4)| = 8
\]
The height from \( O \) to line \( PQ \) (which is vertical) is the horizontal distance from \( O \) to line \( x = 4 \), which is:
\[
4
\]
Thus, the area is:
\[
A = \frac{1}{2} \times 8 \times 4 = 16
\]
### Final Answer
The area of triangle OPQ is \( 16 \).
