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Question

For a positive real number $$k$$ let $$E_{k}$$ be the ellipse with equation
$$\dfrac {x^{2}}{a^{2} + k} + \dfrac {y^{2}}{b^{2} + k} = 1$$
where $$a > b > 0$$. All members of the family of ellipse $$\left \{E_{k} : k > 0\right \}$$ have the same.


A
Foci
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B
Eccentricity
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C
Pair of directrices
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D
Centre
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Solution

The correct options are
B Foci
C Centre
Given ellipse is $$\dfrac { { x }^{ 2 } }{ { a }^{ 2 }+k } +\dfrac { { x }^{ 2 } }{ { b }^{ 2 }+k } =1\\ $$,
Eccentricity of the given ellipse is $${ e }^{ 2 }=\dfrac { { b }^{ 2 }-{ a }^{ 2 } }{ k+{ a }^{ 2 } } $$
This depends on $$k$$,
the focci are $$\pm (ae,0)=\pm (\sqrt { { b }^{ 2 }-{ a }^{ 2 } },0) $$,independent of $$k$$,$$\therefore$$ all the ellipse of this family have the same focci,
centre is always origin $$(0,0)$$.

Maths

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