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

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 given equation of the parabola is:
\[ y^2 + x + 4 = 0 \]
Rearranging gives:
\[ y^2 = -x - 4 \]

2. **Identifying Points on the Parabola:**
Let point A be \( A(0, 2) \). We need to find points B and C on the parabola. We can represent point B as \( B(t_1) = (t_1^2 - 4, t_1) \) and point C as \( C(t_2) = (t_2^2 - 4, t_2) \).

3. **Finding the Slopes:**
The slope of line AB can be calculated as:
\[ m_{AB} = \frac{t_1 - 2}{(t_1^2 - 4) - 0} = \frac{t_1 - 2}{t_1^2 - 4} \]
The slope of line CB is:
\[ m_{CB} = \frac{t_2 - t_1}{(t_2^2 - 4) - (t_1^2 - 4)} = \frac{t_2 - t_1}{t_2^2 - t_1^2} \]

4. **Using the Perpendicular Condition:**
Since \( \angle CBA = \frac{\pi}{2} \), the slopes of lines AB and CB must satisfy:
\[ m_{AB} \cdot m_{CB} = -1 \]
Substituting the expressions for the slopes gives:
\[ \left(\frac{t_1 - 2}{t_1^2 - 4}\right) \cdot \left(\frac{t_2 - t_1}{t_2^2 - t_1^2}\right) = -1 \]

5. **Cross-Multiplying and Rearranging:**
Cross-multiplying leads to:
\[ (t_1 - 2)(t_2 - t_1) = -(t_1^2 - 4)(t_2^2 - t_1^2) \]
This can be simplified to find a relationship between \( t_1 \) and \( t_2 \).

6. **Finding the Range of \( t_2 \):**
To find the range of the ordinate of point C, we need to analyze the conditions derived from the perpendicularity condition. The discriminant of the resulting quadratic equation in \( t_2 \) must be non-negative for real solutions.

7. **Solving the Quadratic Inequality:**
The quadratic inequality derived from the slopes will yield conditions on \( t_2 \). Solving this will give us the range of values for \( t_2 \).

8. **Determining the Range of the Ordinate of C:**
The ordinate of point C is simply \( t_2 \). Based on the analysis, we find that:
\[ t_2 \in (-\infty, 0) \cup [4, \infty) \]

### Final Result:
The range of the ordinate of point C is:
\[ (-\infty, 0) \cup [4, \infty) \]