1.

If `e_(1)ande_(2)` be the eccentricities of the ellipses `(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1and(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1` respectively then prove that `3=4e_(2)^(2)-e_(1)^(2)`.

Answer» For ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Its eccentricity is `e_(1)`
`:." "b^(2)=a^(2)(1-e_(1)^(2))rArr(b^(2))/(a^(2))=1-e_(1)^(2)` . . . (1)
For ellipse `(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1`
`rArr" "(x^(2))/(a^(2))+(4y^(2))/(b^(2)//4)=1`
Its eccentricity is `e_(2)`
`:." "(b^(2))/(4)=a^(2)(1-e_(2)^(2))`
`rArr" "(b^(2))/(4a^(2))=1-e_(2)^(2)" "rArr" "(b^(2))/(a^(2))=4-4e_(2)^(2)` . . . (2)
From eqs. (1) and (2), we get
`4-4e_(2)^(2)=1-e_(1)^(2)`
`rArr" "3=4e_(2)^(2)-e_(1)^(2)`.


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