The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse `x^(2)/9+y^(2)/5=1` is

Options

A. `(2)/(2)`
B. `27`
C. `(27)/(4)`
D. `18`

Answer

B

Explanation

the equation of the ellipse is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1,` where `a^(2)=9, b^(2)=5` Let e be the eccentricity of the ellipse then
`e^(2)=1=-(b^(2))/(a^(2))=1-(5)/(9)=(4)/(9)implies e=(2)/(3)`
the coordinate of the end -point of laturecta are
`(+- ae,+-(b^(2))/(a)) or (+- 2,+-(5)/(3))`
the equation of the tangent at `L_(1) (2,(5)/(3))` is
this meets the coordinate axes at `A(0//2,0) and B(0,3),`
area of quadrilateral `ABCD = 4`( Area of `Delta OAB)`
`=4xx(1)/(2)(OAxxOB)`
`=4xx(1)/(2) xx(9) /(2) xx3=27` sq, units