Latus Rectum is `4 and e=(1)/sqrt(2)` axes are co­ordinate axes, eq. of the ellipse

Options

A. `(x^(2))/(4)+(y^(2))/(8)=1`
B. `(x^(2))/(16)+(y^(2))/(12)=1`
C. `(x^(2))/(16)+(y^(2))/(8)=1`
D. `(x^(2))/(16)+(y^(2))/(4)=1`

Answer

To find the equation of the ellipse given the latus rectum and eccentricity, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Latus Rectum (L) = 4
- Eccentricity (e) = \( \frac{1}{\sqrt{2}} \)

The standard form of the ellipse when the axes are the coordinate axes is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Step 2: Use the formula for the latus rectum
The formula for the latus rectum of an ellipse is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the value of L:
\[ 4 = \frac{2b^2}{a} \]
From this, we can express \( b^2 \) in terms of \( a \):
\[ 2b^2 = 4a \implies b^2 = 2a \]

### Step 3: Use the formula for eccentricity
The eccentricity \( e \) is related to \( a \) and \( b \) by the formula:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting the value of \( e \):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{b^2}{a^2} \]
This simplifies to:
\[ \frac{1}{2} = 1 - \frac{b^2}{a^2} \]
Rearranging gives:
\[ \frac{b^2}{a^2} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 4: Substitute \( b^2 = 2a \) into the eccentricity equation
Substituting \( b^2 = 2a \) into the equation:
\[ \frac{2a}{a^2} = \frac{1}{2} \]
This simplifies to:
\[ \frac{2}{a} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 4 = a \implies a = 4 \]

### Step 5: Calculate \( b^2 \)
Now substituting \( a = 4 \) back into \( b^2 = 2a \):
\[ b^2 = 2 \times 4 = 8 \]

### Step 6: Write the equation of the ellipse
Now we have \( a^2 = 16 \) and \( b^2 = 8 \). Thus, the equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{8} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{8} = 1 \]

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