Suppose the eccentricity of the ellipse `x^2/(a^2+3)+ y^2/(a^2+4)=1` is `1//sqrt8` . Let l be the latus rectum of the ellipse, then l equals

Answer

To solve the problem, we need to find the latus rectum of the ellipse given by the equation:

\[ \frac{x^2}{a^2 + 3} + \frac{y^2}{a^2 + 4} = 1 \]

where the eccentricity \( e \) is given as \( \frac{1}{\sqrt{8}} \).

### Step 1: Identify the semi-major and semi-minor axes
In the standard form of the ellipse, we can identify \( a^2 = a^2 + 4 \) (the larger denominator) and \( b^2 = a^2 + 3 \) (the smaller denominator). Thus, we have:

\[ a^2 = a^2 + 4 \quad \text{and} \quad b^2 = a^2 + 3 \]

### Step 2: Use the formula for eccentricity
The formula for eccentricity \( e \) of an ellipse is given by:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

Substituting the values of \( a^2 \) and \( b^2 \):

\[ e = \sqrt{1 - \frac{a^2 + 3}{a^2 + 4}} \]

### Step 3: Set up the equation with the given eccentricity
We know that \( e = \frac{1}{\sqrt{8}} \). Therefore, we can set up the equation:

\[ \frac{1}{\sqrt{8}} = \sqrt{1 - \frac{a^2 + 3}{a^2 + 4}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:

\[ \frac{1}{8} = 1 - \frac{a^2 + 3}{a^2 + 4} \]

### Step 5: Rearranging the equation
Rearranging the equation, we get:

\[ \frac{a^2 + 3}{a^2 + 4} = 1 - \frac{1}{8} \]

Calculating the right side:

\[ 1 - \frac{1}{8} = \frac{7}{8} \]

So we have:

\[ \frac{a^2 + 3}{a^2 + 4} = \frac{7}{8} \]

### Step 6: Cross-multiply to solve for \( a^2 \)
Cross-multiplying gives:

\[ 8(a^2 + 3) = 7(a^2 + 4) \]

Expanding both sides:

\[ 8a^2 + 24 = 7a^2 + 28 \]

### Step 7: Rearranging to find \( a^2 \)
Rearranging gives:

\[ 8a^2 - 7a^2 = 28 - 24 \]

This simplifies to:

\[ a^2 = 4 \]

### Step 8: Calculate \( b^2 \)
Now substituting \( a^2 \) back to find \( b^2 \):

\[ b^2 = a^2 + 3 = 4 + 3 = 7 \]

### Step 9: Find the latus rectum \( l \)
The formula for the latus rectum \( l \) of an ellipse is given by:

\[ l = \frac{2b^2}{a} \]

Substituting the values we found:

\[ l = \frac{2 \cdot 7}{\sqrt{4}} = \frac{14}{2} = 7 \]

### Final Answer
Thus, the value of the latus rectum \( l \) is:

\[ \boxed{7} \]