If the area of the triangle whose one vertex is at the vertex of the parabola, `y^(2) + 4 (x - a^(2)) = 0` and the other two vertices are the points of intersection of the parabola and Y-axis, is 250 sq units, then a value of 'a' is

Explanation

To solve the problem, we need to find the value of 'a' given the area of the triangle formed by the vertex of the parabola and the points of intersection of the parabola with the Y-axis.

### Step 1: Identify the parabola and its vertex
The given equation of the parabola is:
\[ y^2 + 4(x - a^2) = 0 \]
Rearranging it gives:
\[ y^2 = -4(x - a^2) \]
This represents a parabola that opens to the left. The vertex of the parabola occurs at the point where \( x = a^2 \) and \( y = 0 \). Therefore, the vertex is:
\[ V(a) = (a^2, 0) \]

### Step 2: Find the points of intersection with the Y-axis
To find the points of intersection with the Y-axis, we set \( x = 0 \) in the equation of the parabola:
\[ y^2 = -4(0 - a^2) = 4a^2 \]
Taking the square root gives:
\[ y = 2a \quad \text{and} \quad y = -2a \]
Thus, the points of intersection with the Y-axis are:
\[ P_1(0, 2a) \quad \text{and} \quad P_2(0, -2a) \]

### Step 3: Calculate the area of the triangle
The area \( A \) of a triangle formed by the vertices \( V(a) \), \( P_1 \), and \( P_2 \) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the distance between \( P_1 \) and \( P_2 \) along the Y-axis, which is:
\[ \text{base} = |2a - (-2a)| = |2a + 2a| = 4a \]
The height of the triangle is the horizontal distance from the vertex \( V(a) \) to the Y-axis, which is:
\[ \text{height} = a^2 \]
Thus, the area of the triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4a \times a^2 = 2a^3 \]

### Step 4: Set the area equal to 250 and solve for 'a'
According to the problem, the area of the triangle is given as 250 square units:
\[ 2a^3 = 250 \]
Dividing both sides by 2:
\[ a^3 = 125 \]
Taking the cube root of both sides:
\[ a = 5 \]

### Conclusion
The value of 'a' is:
\[ \boxed{5} \]