Find the equation of an ellipse the distance between the foci is 8 units and the distance between the directrices is 18 units.

Explanation

Let the equation of ellipse be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1andagtb`.
Distance between foci = 2ae = 8
`rArr" "ae=4` . . . (1)
Distance between directrix `=(2a)/(e)=18`
`rArr" "(a)/(3)=9` . . . (2)
Multiplying eqs. (1) and (2)
`ae*(a)/(e)=4*9`
`rArr""a^(2)=336`
and `b^(2)=a^(2)(1-e^(2))`
`=a^(2)-(ae)^(2)`
`=36-(4)^(2)=20`
Therefore, the equation of ellipse
`(x^(2))/(36)+(y^(2))/(20)=1`.