Given f is increasing, the equation `x^(2)/(f(2a))+y^(2)/(f(a^(2)-3))=1` represents an ellipse with X-axis as major axis if

Options

A. [-1,3]
B. [1,3]
C. (-1,3)
D. (0,5)

Answer

To determine the conditions under which the given equation represents an ellipse with the x-axis as the major axis, we can follow these steps:

### Step 1: Understand the given equation
The equation given is:

\[ \frac{x^2}{f(2a)} + \frac{y^2}{f(a^2 - 3)} = 1 \]

This is in the standard form of an ellipse:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \( a^2 = f(2a) \) and \( b^2 = f(a^2 - 3) \).

### Step 2: Identify the condition for the major axis
For the ellipse to have the x-axis as the major axis, we need:

\[ a^2 > b^2 \]

This translates to:

\[ f(2a) > f(a^2 - 3) \]

### Step 3: Use the property of the increasing function
Since \( f \) is given to be an increasing function, we can infer that if \( f(2a) > f(a^2 - 3) \), then:

\[ 2a > a^2 - 3 \]

### Step 4: Rearrange the inequality
Rearranging the inequality gives:

\[ a^2 - 2a - 3 < 0 \]

### Step 5: Factor the quadratic expression
Factoring the quadratic expression, we get:

\[ (a + 1)(a - 3) < 0 \]

### Step 6: Determine the intervals
To find the intervals where this inequality holds, we can analyze the sign changes around the roots \( a = -1 \) and \( a = 3 \).

- The critical points divide the number line into intervals: \( (-\infty, -1) \), \( (-1, 3) \), and \( (3, \infty) \).
- Testing a point from each interval:
- For \( a = -2 \) (in \( (-\infty, -1) \)): \( (-2 + 1)(-2 - 3) = (-1)(-5) > 0 \)
- For \( a = 0 \) (in \( (-1, 3) \)): \( (0 + 1)(0 - 3) = (1)(-3) < 0 \)
- For \( a = 4 \) (in \( (3, \infty) \)): \( (4 + 1)(4 - 3) = (5)(1) > 0 \)

Thus, the inequality \( (a + 1)(a - 3) < 0 \) holds in the interval:

\[ a \in (-1, 3) \]

### Final Answer
The condition for the equation to represent an ellipse with the x-axis as the major axis is:

\[ a \in (-1, 3) \]