An ellipse circumscribes a quadrilateral whose sides are given by `x=-+2 and y =+-4`. If the distance between focis is `4sqrt(6)` and major axis is along yaxis then find the eqution of ellipse

Answer

`(x^(2))/(8)+(y^(2))/(32)=1`

Explanation

Let the required ellipse be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
where `a^(2)=b^(2)(1-e^(2))`.
Ellipse passes through the point (2,4)
`:. (4)/(a^(2))+(16)/(b^(2))`
Also, distance between foci is `4sqrt(6)`.

`:. be=2sqrt(6)`
`:. (4)/(b^(2)-24)+(16)/(b^(2))=1`
`rArrr b^(4)-44b^(2)+16xx24=0`
`rArr(b^(2)-32)(b^(2)-12)=0`
`:. b^(2)=32(as bgt4)`
and `a^(2)=8`
therefore, the requiredis `(x^(2))/(8)+(y^(2))/(32)=1`