Let `A (0,2),B` and C be points on parabola `y^(2)+x +4` such that `/_CBA (pi)/(2)`. Then the range of ordinate of C is (a) `(-oo,0)uu (4,oo)` (b) `(-oo,0] uu[4,oo)` (c) `[0,4]` (d) `(-oo,0)uu [4,oo)`

Options

A. `(-oo,0)uu (4,oo)`
B. `(-oo,0] uu[4,oo)`
C. `[0,4]`
D. `(-oo,0)uu [4,oo)`

Explanation

To solve the problem, we need to find the range of the ordinate (y-coordinate) of point C on the parabola defined by the equation \( y^2 + x + 4 = 0 \) given that the angle \( \angle CBA = \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Rearranging the Parabola Equation**:
The equation of the parabola can be rearranged to express \( x \) in terms of \( y \):
\[ x = -y^2 - 4 \]

2. **Identifying Points A, B, and C**:
- Point \( A \) is given as \( A(0, 2) \).
- Let point \( B \) be \( B(t_1^2 - 4, t_1) \) and point \( C \) be \( C(t_2^2 - 4, t_2) \), where both points lie on the parabola.

3. **Finding Slopes**:
- The slope of line segment \( AB \) is given by:
\[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{t_1 - 2}{t_1^2 - 4 - 0} = \frac{t_1 - 2}{t_1^2 - 4} \]
- The slope of line segment \( CB \) is given by:
\[ m_{CB} = \frac{y_C - y_B}{x_C - x_B} = \frac{t_2 - t_1}{t_2^2 - 4 - (t_1^2 - 4)} = \frac{t_2 - t_1}{t_2^2 - t_1^2} \]

4. **Condition for Perpendicularity**:
Since \( \angle CBA = \frac{\pi}{2} \), the slopes must satisfy:
\[ m_{AB} \cdot m_{CB} = -1 \]
This leads to:
\[ \left(\frac{t_1 - 2}{t_1^2 - 4}\right) \cdot \left(\frac{t_2 - t_1}{t_2^2 - t_1^2}\right) = -1 \]

5. **Solving the Equation**:
Rearranging and simplifying the equation leads to a quadratic equation in terms of \( t_1 \) and \( t_2 \). After solving, we find:
\[ t_1^2 + 2t_2 + 1 = 0 \]
This indicates a relationship between \( t_1 \) and \( t_2 \).

6. **Finding the Range of \( t_2 \)**:
The discriminant of the quadratic must be non-negative for real solutions:
\[ 2^2 - 4(1)(-t_2) \geq 0 \implies 4 + 4t_2 \geq 0 \implies t_2 \geq -1 \]
However, since \( t_2 \) represents the ordinate of point \( C \) on the parabola, we also need to consider the shape of the parabola.

7. **Final Range for Ordinate of C**:
The parabola opens to the left, and thus the y-values can be:
\[ y_C \in (-\infty, 0) \cup [4, \infty) \]

### Conclusion:
The range of the ordinate of point C is:
\[ (-\infty, 0) \cup [4, \infty) \]
Thus, the correct option is **(d) \((-∞, 0) \cup [4, ∞)\)**.