A vertical line passing through the point `(h, 0)` intersects the ellipse `x^2/4+y^2/3=1` at the points `P` and `Q`.Let the tangents to the ellipse at P and Q meet at `R`. If `Delta (h)` Area of triangle `DeltaPQR`, and `Delta_1 = max_(1/2<=h<=1)Delta(h)` and `Delta_2 = min_(1/2<=h<=1) Delta (h)` Then `8/sqrt5 Delta_1-8Delta_2`

Explanation

`(x^(2))/(4)+(y^(2))/(3)=1`
`:. Y=(sqrt(3))/(2)sqrt(4-h^(2)) "at"x=h`
Let `R(x_(1),0)`
PQ, is the chord of contant So, `(x x_(1))/(4)=1`
or `x=(4)/(x_(1))`
Which is the euation of PQ. At x=h
`(x)/(x_(1))= h or x_(1)=(4)/(h)`
`Delta(h)="Area of" DeltaPQR=(1)/(2)xxPQxxRT`
`=(1)/(2)xx(2sqrt(3))/(2)sqrt(4-h^(2))xx(x_(1)-h)=(sqrt(3))/(2h)(4-h^(2))^(3//2)`
Clearly, `Deltah=(45sqrt(5))/(8)"at" h=(1)/(2)`
and `Delta_(2)` Minimum of `Delta(h)=(9)/(2)"ath=1`
So, `(8)/(sqrt(5))Delta_(1)-8Delta_(2)=(8)/(sqrt(5))xx(45sqrt(5))/(8)-xx(9)/(2)=45-36=9`