If O A =2 i +2 j + k , O B =2 i +4 j +4 k and the length of the internal bisector of B O A of triangle A O B is k , then 9 k^2=

Step-by-step solution

Given vectors aligned & O A & =2 i +2 j + k and & O B & =2 i +4 j +4 k & | O A | & = 4+4+1 =3=m (let) and & | O B | & = 4+16+16 =6=n (let) aligned and The angle bisector of B O A intersect the side A B at point P in the ratio 3: 6=1: 2 , so aligned O P & = 2( O A )+1( O B ) 3 & = 6 i +8 j +6 k 3 =2 i + 8 3 j +2 k | O P | & = 2^2+ ( 8 3 )^2+2^2 & = 4+ 64 9 +4 & = 72+64 9 = 136 9 =k 9 k^2 & =9 ( 136 9 )=136 aligned