The equation `(x^(2))/(14-a)+(y^(2))/(9-a)=1` represent

Options

A. an ellipse if `a lt 9`
B. a hyperbola ifi `9 lt a lt 14`
C. a hyperbola if `agt14`
D. an ellipse `agt9`

Answer

To determine the type of conic section represented by the equation

\[ \frac{x^2}{14-a} + \frac{y^2}{9-a} = 1, \]

we need to analyze the conditions under which this equation represents an ellipse, a hyperbola, or a parabola.

### Step 1: Identify the general form
The given equation is in the form

\[ \frac{x^2}{A} + \frac{y^2}{B} = 1, \]

where \( A = 14 - a \) and \( B = 9 - a \).

### Step 2: Determine the conditions for \( A \) and \( B \)
For the equation to represent a conic section, both \( A \) and \( B \) must be positive. Therefore, we need:

1. \( 14 - a > 0 \)
2. \( 9 - a > 0 \)

### Step 3: Solve the inequalities
From the first inequality:

\[ 14 - a > 0 \implies a < 14. \]

From the second inequality:

\[ 9 - a > 0 \implies a < 9. \]

### Step 4: Combine the conditions
The more restrictive condition is \( a < 9 \). Therefore, for the equation to represent a valid conic section, we need:

\[ a < 9. \]

### Step 5: Determine the type of conic section
Now, we need to analyze the type of conic section based on the signs of \( A \) and \( B \):

- If both \( A \) and \( B \) are positive, the equation represents an ellipse.
- If \( A \) is positive and \( B \) is negative, it represents a hyperbola.
- If either \( A \) or \( B \) is zero, it represents a parabola.

Since we have established that \( a < 9 \), we also need to check when \( 9 - a \) becomes negative:

1. If \( a < 9 \), then \( 9 - a > 0 \) (which means \( B > 0 \)).
2. If \( a \geq 9 \), then \( 9 - a \leq 0 \) (which means \( B \leq 0 \)).

Thus, for \( a < 9 \), both \( A \) and \( B \) are positive, and the equation represents an ellipse.

### Step 6: Check for hyperbola conditions
Next, we check if there are any values of \( a \) that could lead to a hyperbola. For a hyperbola, we would need \( A > 0 \) and \( B < 0 \):

1. \( 14 - a > 0 \implies a < 14 \).
2. \( 9 - a < 0 \implies a > 9 \).

Thus, for a hyperbola, \( a \) must satisfy \( 9 < a < 14 \).

### Conclusion
In summary, the equation

\[ \frac{x^2}{14-a} + \frac{y^2}{9-a} = 1 \]

represents:
- An ellipse if \( a < 9 \).
- A hyperbola if \( 9 < a < 14 \).