A vertical line passing through the point (h, 0) intersects the ellipse `(x^(2))/(4)+(y^(2))/(3)=1` at the point P and Q. Let the tangents to the ellipse at P and Q meet at the point R.
If `Delta(h)= " Area of the "DeltaPQR, Delta_(1)(h)=underset(1//2lehle1)max Delta(h)`
and `Delta_(2)(h)=underset(1//2lehle1)minDelta(h), then(8)/(sqrt5)Delta_(1)-8Delta_(2)=`

Options

A. `(36)/(8)`
B. `(45sqrt5)/(8)`
C. 9
D. 8

Answer

D

Explanation

Let the coordinates of P be `(2costheta, sqrt3sintheta)`. Then, the coordinates of Q are `(2costheta, -sqrt3sintheta)`. Then `h = 2costheta`.
The equation of tangents at P and Q are `x//2 costheta+1sqrt3ysintheta=1 and, x//2costheta-y//sqrt3sintheta=1`. These two intersect at R `(2 sec theta, 0)`

`therefore Delta(h)=(1)/(2)(2sectheta-h)(2sqrt3sintheta)[thereforePQ=2sqrt3sinthetafand MR=(2sectheta-h)]`
`rArr Delta(h)=sqrt3(2sectheta-2costheta)sintheta`
`rArr Delta(h)=2sqrt3(sin^(3)theta)/(costheta)" "[thereforecostheta=(h)/(2)]`
`Delta(h)=(4sqrt3)/(h)(1-(h^(2))/(4))=(sqrt3)/(2)((4-h^(2))^(3//2))/(h)`
`(d)/(dh)(Delta(h))=(-sqrt3(2+h^(2))sqrt(4-h^(2)))/(h^(2))lt0 " for "hin(-2,2)`
`rArr Delta(h)` is a decreasing function of h on (-2, 2).
`Delta_(1)=underset(1//2lehle1)maxDelta(h)=Delta((1)/(2))=(45sqrt5)/(8)`
`Delta_(2)=underset(1//2lehle1)minDelta(h)=Delta(1)=(9)/(2)`
Hence, `(8)/sqrt5 Delta_(1)-8Delta_(2)=45-36=9`.