If length of the major axis is 8 and e = `1/sqrt(2)`Axes are co-ordinate axes then equation of the ellipse is

Options

A. `(x^(2))/(12)+(y^(2))/(4)=1`
B. `(x^(2))/(16)+(y^(2))/(8)=1`
C. `(x^(2))/(24)+(y^(2))/(16)=1`
D. `(x^(2))/(32)+(y^(2))/(24)=1`

Answer

To find the equation of the ellipse given the length of the major axis and the eccentricity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the length of the major axis**:
The length of the major axis is given as 8. In an ellipse, the length of the major axis is equal to \(2a\), where \(a\) is the semi-major axis.
\[ 2a = 8 \implies a = \frac{8}{2} = 4 \]

2. **Calculate \(a^2\)**:
Now, we calculate \(a^2\):
\[ a^2 = 4^2 = 16 \]

3. **Use the eccentricity formula**:
The eccentricity \(e\) of the ellipse is given as \(\frac{1}{\sqrt{2}}\). The relationship between the semi-major axis \(a\), semi-minor axis \(b\), and eccentricity \(e\) is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Squaring both sides gives:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting \(e = \frac{1}{\sqrt{2}}\):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{b^2}{16} \]
This simplifies to:
\[ \frac{1}{2} = 1 - \frac{b^2}{16} \]

4. **Rearranging the equation**:
Rearranging the equation to solve for \(b^2\):
\[ \frac{b^2}{16} = 1 - \frac{1}{2} = \frac{1}{2} \]
Multiplying both sides by 16 gives:
\[ b^2 = 16 \cdot \frac{1}{2} = 8 \]

5. **Write the equation of the ellipse**:
The standard form of the equation of an ellipse with the major axis along the x-axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting \(a^2 = 16\) and \(b^2 = 8\):
\[ \frac{x^2}{16} + \frac{y^2}{8} = 1 \]

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