If the chord, joining two points whose eccentric angles are `alpha and beta`, cuts the major axis ofthe ellipse `x^2/a^2 + y^2/b^2=1` at a distance c from the centre, then `tan alpha//2.tan beta//2` is equal to

Options

A. `(c+a)/(c-a)`
B. `(c-a)/(c+a)`
C. `(a-c)/(a+c)`
D. `(a+c)/(a-c)`

Answer

B

Explanation

the equation of the chord joining points whose eccentric angle are `alpha and beta` on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`, is
`(x)/(a) cos ((alpha+beta)/(2))+(Y)/(b) sin ""((alpha+beta)/(2))= cos ""((alpha-beta)/(2))`
this will cut the major axis at the point (c,0) ,if
`(c)/(a) cos ((alpha+beta)/(2) )= cos ((alpha - beta)/(2))`
`implies ( cos ((alpha+beta)/(2)))/(cos ((alpha- beta)/(2)))=(a)/(c)`
`implies (cos ((alpha+beta)/(2))+ cos ((alpha-beta)/(2)))/(cos ((alpha+beta)/(2))- cos ((alpha- beta)/(2)))=(c-a)/(c+a)`
`implies (2 cos alpha //2cos beta //2)/(- 2 sin alpha //2 sin beta //2 )=-(a+c)/(a-c)implies tan ""(alpha)/(2) tan (beta)/(2) =(c-a)/(c+a)`