The length of the latusrectum of an ellipse is `(18)/(5)` and eccentricity is `(4)/(5)`, then equation of the ellipse is . .

Options

A. `(x^(2))/(25)+(y^(2))/(8)=1`
B. `(x^(2))/(25)+(y^(2))/(9)=1`
C. `(x^(2))/(25)+(y^(2))/(16)=1`
D. `(x^(2))/(16)+(y^(2))/(9)=1`

Answer

B

Explanation

Given `(2b^(2))/(a)=(18)/(5)implies(b^(2))/(a)=(9)/(5)`
`impliesb^(2)=(9)/(5)a` . . . (i)
and, `e=sqrt(1-(b^(2))/(a^(2)))=(4)/(5)implies1-(b^(2))/(a^(2))=(16)/(25)`
`implies(b^(2))/(a^(2))=1-(16)/(25)=(9)/(25)`
`implies(((9)/(5))a)/(a^(2))=(9)/(25)implies(9)/(5a)=(9)/(25)`
`thereforeb^(2)=(9)/(5)(5)=9` [using Eq. (i)]
`therefore` Required equation of ellipse,
`(x^(2))/(25)+(y^(2))/(9)=1`