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The area (in sq. units) of the quadril...

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 (a) `27/4` (b) `18` (c) `27/2` (d) `27`

A

`27//4`

B

18

C

`27//2`

D

27

Text Solution

Verified by Experts

The correct Answer is:
D
The given ellipe is
`(x^(2))/(9)+(y^(2))/(5)=1`

Then `a^(2)=9,b^(2)=5`. Therefore,
`e=sqrt(1-(5)/(9))=(2)/(3) " " :. ae=2`
Hence, the endpoint of latus rectum in the first quadrant is `L(ae,b^(2)//a)-=L(2,5//3)`
The equation of tangent at L is `(2x)/(9)+(y)/(3)=1`
the tangents meets the x-axis at A(9/2,0) and the y-axis at B(0,3). Therefore,
Area of `DeltaOAPB=(1)/(2)xx(9)/(2)xx3=(27)/(4)`
`:.` Area of quadrilateral `=4xx(27)/(4)=27`
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