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If a,b,c and A,B,C inR-{0} such that aA+...

If a,b,c and A,B,C `in`R-{0} such that `aA+bB+cD+ sqrt((a^(2)+b^(2)+c^(2))(A^(2)+B^(2)+C^(2)))=0`, then value of `(aB)/(bA) +(bC)/(cB) + (cA)/(aC)` is

A

3

B

4

C

5

D

6

Text Solution

Verified by Experts

The correct Answer is:
A
Let `theta` be the required angle. Then `theta` will be the angle between `veca` and `vecb + vecc(vecb+vecc)` lies along the angular bisector of `veca` and `vecb`.
`therefore costheta=(veca.(vecb+vecc))/(|veca||vecb+vecc|)`
`=(2cosalpha)/sqrt(2+2cosalpha) = (cosalpha)/(cosalpha/2)`
`therefore theta=cos^(-1)(cosalpha)/(cosalpha/2)`
`therefore vecr_(1)vecr_(2)-|vecr_(1)||vecr_(2)|`
`rArr vecr_(1)` and `vecr_(2)` are anti-parallel
`rArr a/A=b/B=c/C=k` where k is any constant
`rArr (aB)/(bA) +(bC)/(cB) + (cA)/(aC)=3`
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