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The line joining $A (b \cos a, b \sin a)$ and $B (a \cos b, a \sin b)$ is produced to the point $M (x, y)$ so that $A M : M B = b : a,$ , then $x \cos \frac{α + β}{2} + y \sin \frac{α + β}{2} =$

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a

-1

b

0

c

1

d

a2+b2

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detailed solution

Correct option is B

As M divides AB externally in the ratio b : ax=b(acos⁡β)−a(bcos⁡α)b−a and y=b(asin⁡β)−a(bsin⁡α)b−a⇒ xy=cos⁡β−cos⁡αsin⁡β−sin⁡α=2sin⁡α+β2sin⁡α−β22cos⁡α+β2sin⁡β−α2⇒ xcos⁡α+β2+ysin⁡α+β2=0

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