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Find the area of the ellipse (x^2)/(a^2)...

Find the area of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`.

Text Solution

Verified by Experts

The correct Answer is:
A, B
given , `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Foci `F_(1) and f_(2)` are `(-ae,0) ` and (ae-0) respectively .Let p(x,y) be any variable point on the ellipse .
the area A of triangle `PF_(1)F_(2)` is given by

`A=(1)/(2) |{:(x,y,1),(-ae,0,1),(ae,0,1):}|`
`-(1)/(2)(-y)(-aexx1-aexx1)`
`=- (1)/(2) y-(-2ae)=a ey = ae.bsqrt(1-(x^(2))/(a^(2)))`
so A is maximum when x=0
`therefore ` Maximum of A `=abc=absqrt(1-(b^(2))/(a^(2))=absqrt(a^(3)-b^(2))/(a^(2))`
`= bsqrt(a^(2)-b^(2))`
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