If `9-x^2>|x-a |` has atleast one negative solution, where `a in R` then complete set of values of a is

Options

A. `(-25/2,2)`
B. `(-35/4,1)`
C. `(-37/2, 0)`
D. `(-37/4, 9)`

Answer

To solve the inequality \( 9 - x^2 > |x - a| \) for at least one negative solution, we will analyze the conditions under which this inequality holds true.

### Step-by-Step Solution:

1. **Understanding the Inequality**:
We need to analyze the inequality \( 9 - x^2 > |x - a| \). This means that the left-hand side must be greater than the right-hand side for at least one negative value of \( x \).

2. **Finding the Range of \( 9 - x^2 \)**:
The expression \( 9 - x^2 \) is a downward-opening parabola with its vertex at \( (0, 9) \). The maximum value occurs at \( x = 0 \) and is equal to 9. As \( x \) moves away from 0, \( 9 - x^2 \) decreases. The parabola intersects the x-axis at \( x = -3 \) and \( x = 3 \).

3. **Analyzing the Right-Hand Side**:
The expression \( |x - a| \) represents the distance between \( x \) and \( a \). For negative values of \( x \), we can rewrite it as:
\[ |x - a| = a - x \quad \text{(since \( x < a \))} \]

4. **Setting Up the Inequality**:
For \( x < 0 \), we need:
\[ 9 - x^2 > a - x \]
Rearranging gives:
\[ 9 + x - x^2 > a \]
This means we need to find the maximum value of \( 9 + x - x^2 \) for negative \( x \).

5. **Finding the Maximum Value**:
The function \( f(x) = 9 + x - x^2 \) is a downward-opening parabola. To find its maximum, we can complete the square or use calculus. The vertex \( x \) of a parabola \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 1 \):
\[ x = -\frac{1}{2(-1)} = \frac{1}{2} \]
Since we are interested in negative \( x \), we check the endpoints \( x = 0 \) and \( x = -3 \):
- At \( x = 0 \): \( f(0) = 9 \)
- At \( x = -3 \): \( f(-3) = 9 - 9 = 0 \)

The maximum value occurs at \( x = 0 \) and is \( 9 \).

6. **Setting the Condition for \( a \)**:
For the inequality \( 9 + x - x^2 > a \) to hold for at least one negative \( x \), we require:
\[ a < 9 \]

7. **Finding the Minimum Value**:
We also need to ensure that \( 9 + x - x^2 \) is positive for some negative \( x \). The minimum value occurs at \( x = -3 \), where \( f(-3) = 0 \). Thus, we need:
\[ a > -\frac{37}{4} \]

8. **Final Set of Values for \( a \)**:
Combining the two conditions, we find:
\[ -\frac{37}{4} < a < 9 \]

### Conclusion:
The complete set of values for \( a \) is:
\[ \boxed{(-\frac{37}{4}, 9)} \]