Q:-Let the points A9a,b,c) and B(d,e,f) be at distances p and q from origin. The line AB passes through origin whenA. (d/a)=(e/b)=(f/c)B. ad+be+cf=pqC. ad+be+cf=p2+q2D. None of these

Dear Tapasranjan

AB line passes through origin so slope of line joining the origin and A  , and slope of line joining the origin and B will be same

d-0 /a-0   = e-0 /b-0   = f-0 /c-0

or

 d/a = e/b  =f/c = k   .....................1

option a is correct

now use distance formula

a² +b²+ c² = p²   

and

d² + e² + f² =q²

now

 put value of d,e,f in terms of a,b,c

 k² ( a² +b²+ c² ) = q² 

 k² ( p² ) = q²  

 k = q/p

now let  ad +eb + fc

          = k(a² +b²+ c² )

          =q/p  ( p²)

          =pq

  option b is also correct

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