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For the ellipse (x^(2))/(a^(2))+(y^(2))/...

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

A

The foci of each ellipse always lie within the other ellipse

B

Their auxiliary circles are the same

C

Their director circles are the same

D

The ellipses encloses the same area

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
Without loss of generality assume `a gt b` Foci of `1^(st)` ellipse are `(+- ae, 0)`
Putting this point in `(x^(2))/(b^(2)) + (y^(2))/(a^(2)) -1`, we get `(a^(2)e^(2))/(b^(2)) -1`
The above quantity may be negative or positive, hence option (a) is not correct
Auxiliary circle for `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1` is `x^(2) + y^(2) =a^(2)` for `(x^(2))/(b^(2)) + (y^(2))/(a^(2)) =1` is `x^(2) + y^(2) =a^(2)`
Director circle for `(x^(2))/(b^(2)) +(y^(2))/(a^(2)) =1` is `x^(2)+y^(2) = a^(2) +b^(2)`
`(x^(2))/(b^(2)) +(y^(2))/(a^(2)) =1` is
Area of the ellipse `= pi ab`
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