The line joining `A(bcosalpha,bsinalpha)` and `B(acosbeta,asinbeta)` is produced to the point `M(x ,y)` so that `A M` and `B M` are in the ratio `b : adot` Then`xcos((alpha+beta)/2)+ysin((alpha+beta)/2)`

Options

A. (-1)
B. 0
C. 1
D. `a^(2)+b^(2)`

Answer

B

Explanation

Clearly , M divides AB expternally in the ratio `b:a`.
`:. x=(b(acos beta)-a(b cos alpha))/(b-a) and,y=(b(a sin beta)-a(b sin alpha))/(b-a)`
`rArr(x)/(y)=(cos beta-cos alpha)/(sin beta- sin alpha)`
`rArr (x)/(y)(2 "sin" (alpha+beta)/(2) "sin"(alpha-beta)/(2))/(-2 "sin" (alpha+beta)/(2) "sin"(alpha-beta)/(2))`
`rArr x "cos" ((alpha+beta)/(2)) + y "sin"((alpha+beta)/(2))=0`