For the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1` and `(x^(2))/(b^(2))+(y^(2))/(a^(2)) =1`

Options

A. (a) The foci of each ellipse always lie within the other ellipse
B. (b) Their auxiliary circles are the same
C. (c) Their director circles are the same
D. (d) The ellipses encloses the same area

Explanation

To solve the problem regarding the two ellipses given by the equations \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), we will analyze each option provided in the question step by step.

### Step 1: Identify the Foci of Each Ellipse

1. **First Ellipse**: The foci of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b\)) are given by the formula \((\pm ae, 0)\), where \(e = \sqrt{1 - \frac{b^2}{a^2}}\).
2. **Second Ellipse**: The foci of the ellipse \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) (where \(a > b\)) are given by \((\pm be', 0)\), where \(e' = \sqrt{1 - \frac{a^2}{b^2}}\).

**Conclusion**: The foci of the first ellipse are always on the x-axis, while the foci of the second ellipse can be either inside or outside depending on the values of \(a\) and \(b\). Therefore, the first option is **incorrect**.

### Step 2: Determine the Auxiliary Circles

1. **Auxiliary Circle of First Ellipse**: The auxiliary circle for the first ellipse is given by the equation \(x^2 + y^2 = a^2\).
2. **Auxiliary Circle of Second Ellipse**: The auxiliary circle for the second ellipse is also given by the equation \(x^2 + y^2 = a^2\).

**Conclusion**: Since both ellipses have the same auxiliary circle, the second option is **correct**.

### Step 3: Find the Director Circles

1. **Director Circle of First Ellipse**: The equation of the director circle for the first ellipse is \(x^2 + y^2 = a^2 + b^2\).
2. **Director Circle of Second Ellipse**: The equation of the director circle for the second ellipse is also \(x^2 + y^2 = a^2 + b^2\).

**Conclusion**: Both ellipses have the same director circle, so the third option is **correct**.

### Step 4: Calculate the Area of Each Ellipse

1. **Area of First Ellipse**: The area of the first ellipse is given by the formula \(A_1 = \pi ab\).
2. **Area of Second Ellipse**: The area of the second ellipse is also given by the formula \(A_2 = \pi ab\).

**Conclusion**: Both ellipses enclose the same area, thus the fourth option is **correct**.

### Final Conclusion

The correct options are:
- Option B (Auxiliary circles are the same)
- Option C (Director circles are the same)
- Option D (Ellipses enclose the same area)