If the chord through the points whose eccentric angles are `theta` and `varphi` on the ellipse `(x^2)/(25)+(y^2)/9=1` passes through a focus, then the value of `tan(theta/2)tan(varphi/2)` is `1/9` (b) `-9` (c) `-1/9` (d) 9

Options

A. `1//9`
B. `-9`
C. `-1//9`
D. 9

Explanation

The equation of the line joining `theta and phi` is
`(x)/(5)cos ((theta+phi)/(2))(y)/(3) sin ((theta+phi)/(2))= cos (theta-phi)/(2))`
If it passes through the point (4,0) then
`(4)/(5) cos ((theta+phi)/(2)) = cos (theta-phi)/(2))`
or `(4)/(5)=(cos {(theta-phi)//2})/(cos {(theta+phi)//2)})`
`or (4+5)/(4-5)=(cos {(theta-phi)//2}+cos {(theta+phi)//2})/(cos {(theta-phi)//2}-cos {{(theta+phi)//2}))`
`=(2cos(theta//2)cos (phi//2))/(2 sin (phi//2)sin (phi//2))`
or `tan.( theta)/(2) tan.(phi)/(2)=-(1)/(9)`
If it passes through the point (-4,0) then `tan.(phi)/(2)tan.(theta)/(2)=9`