Consider the ellipses `E_1:x^2/9+y^2/5=1` and `E_2: x^2/5+y^2/9=1` . Both the ellipses have

Options

A. the same foci
B. same major axis
C. the same minor axis
D. the same eccentricity

Answer

To solve the problem involving the two ellipses \( E_1: \frac{x^2}{9} + \frac{y^2}{5} = 1 \) and \( E_2: \frac{x^2}{5} + \frac{y^2}{9} = 1 \), we will analyze their properties step by step.

### Step 1: Identify the parameters of the ellipses

For the ellipse \( E_1: \frac{x^2}{9} + \frac{y^2}{5} = 1 \):
- Here, \( a^2 = 9 \) and \( b^2 = 5 \).
- Thus, \( a = 3 \) and \( b = \sqrt{5} \).

For the ellipse \( E_2: \frac{x^2}{5} + \frac{y^2}{9} = 1 \):
- Here, \( a^2 = 9 \) and \( b^2 = 5 \) (note that the roles of \( a \) and \( b \) are switched).
- Thus, \( a = 3 \) and \( b = \sqrt{5} \).

### Step 2: Determine the orientation of the major and minor axes

For \( E_1 \):
- Since \( a^2 > b^2 \), the major axis is along the x-axis and the minor axis is along the y-axis.

For \( E_2 \):
- Since \( a^2 > b^2 \) (with \( a \) still being 3), the major axis is along the y-axis and the minor axis is along the x-axis.

### Step 3: Calculate the foci of both ellipses

The foci of an ellipse are given by the formula:
\[ c = \sqrt{a^2 - b^2} \]

For \( E_1 \):
- \( c = \sqrt{9 - 5} = \sqrt{4} = 2 \).
- The foci are located at \( (±c, 0) = (±2, 0) \).

For \( E_2 \):
- \( c = \sqrt{9 - 5} = \sqrt{4} = 2 \).
- The foci are located at \( (0, ±c) = (0, ±2) \).

### Step 4: Calculate the eccentricity of both ellipses

The eccentricity \( e \) of an ellipse is given by:
\[ e = \frac{c}{a} \]

For both ellipses:
- \( e = \frac{2}{3} \).

### Conclusion

1. **Foci**:
- \( E_1 \) has foci at \( (2, 0) \) and \( (-2, 0) \).
- \( E_2 \) has foci at \( (0, 2) \) and \( (0, -2) \).

2. **Eccentricity**:
- Both ellipses have the same eccentricity \( e = \frac{2}{3} \).

3. **Major and Minor Axes**:
- \( E_1 \) has its major axis along the x-axis and minor axis along the y-axis.
- \( E_2 \) has its major axis along the y-axis and minor axis along the x-axis.