If eccentricity of the ellipse `(x^(2))/(a^(2)+1)+(y^(2))/(a^(2)+2)=1` is `(1)/(sqrt6)`, then the ratio of the length of the latus rectum to the length of the major axis is

Options

A. `(5)/(6)`
B. `(3)/(sqrt6)`
C. `(2)/(3)`
D. `(2)/(sqrt6)`

Answer

To solve the problem, we need to find the ratio of the length of the latus rectum to the length of the major axis of the given ellipse. Let's break this down step by step.

### Step 1: Identify the given ellipse equation
The equation of the ellipse is given as:
\[ \frac{x^2}{a^2 + 1} + \frac{y^2}{a^2 + 2} = 1 \]
From this equation, we can identify the semi-major axis \( b^2 = a^2 + 2 \) and the semi-minor axis \( a^2 = a^2 + 1 \).

### Step 2: Determine the eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Here, we know that \( e = \frac{1}{\sqrt{6}} \).

### Step 3: Set up the equation for eccentricity
Substituting the values of \( a^2 \) and \( b^2 \) into the eccentricity formula:
\[ \frac{1}{\sqrt{6}} = \sqrt{1 - \frac{a^2 + 2}{a^2 + 1}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{1}{6} = 1 - \frac{a^2 + 2}{a^2 + 1} \]

### Step 5: Rearranging the equation
Rearranging the equation leads to:
\[ \frac{a^2 + 2}{a^2 + 1} = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 6: Cross-multiply to solve for \( a^2 \)
Cross-multiplying gives:
\[ 6(a^2 + 2) = 5(a^2 + 1) \]
Expanding both sides results in:
\[ 6a^2 + 12 = 5a^2 + 5 \]
Rearranging gives:
\[ a^2 = 7 \]

### Step 7: Calculate \( b^2 \)
Now substituting \( a^2 \) back to find \( b^2 \):
\[ b^2 = a^2 + 2 = 7 + 2 = 9 \]

### Step 8: Calculate the lengths of the latus rectum and the major axis
The length of the latus rectum \( L \) is given by:
\[ L = \frac{2b^2}{a} \]
The length of the major axis \( A \) is given by:
\[ A = 2a \]
Substituting the values:
\[ L = \frac{2 \times 9}{\sqrt{7}} = \frac{18}{\sqrt{7}} \]
\[ A = 2\sqrt{7} \]

### Step 9: Calculate the ratio of the lengths
Now we need to find the ratio of the length of the latus rectum to the length of the major axis:
\[ \text{Ratio} = \frac{L}{A} = \frac{\frac{18}{\sqrt{7}}}{2\sqrt{7}} = \frac{18}{2 \times 7} = \frac{18}{14} = \frac{9}{7} \]

### Final Answer
Thus, the ratio of the length of the latus rectum to the length of the major axis is:
\[ \frac{9}{7} \]