Let A, B, C be three points in x y -plane, whose position vector are given by 3 i ^ + j ^ , i ^ + 3 j ^ and a i ^ + ( 1 − a ) j ^ respectively with respect to the origin O . If the distance

Question

Let A, B, C be three points in x y -plane, whose position vector are given by 3 ​ i ^ + j ^ ​ , i ^ + 3 ​ j ^ ​ and a i ^ + ( 1 − a ) j ^ ​ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors OA and OB is 2 ​ 9 ​ , then the sum of all the possible values of a is :

TestHub · Accepted Answer

Equation of line in the internal bisector of O A ˙ and OB is ( 3 ​ + 1 ) i ^ + ( 3 ​ + 1 ) j ^ ​ ⇒ line will be y = x ⇒ x − y = 0 D = ​ a 2 + ( 1 − a ) 2 ​ a − ( 1 − a ) ​ ​ = 2 ​ 9 ​ ​ ( 2 a − 1 ) 2 = 2 81 ​ ( a 2 + ( 1 − a ) 2 ) ⇒ 2 ( 4 a 2 − 4 a + 1 ) = 81 a 2 + 81 a 2 − 162 a − 81 ⇒ 162 a 2 − 162 a + 81 − 8 a 2 + 8 a − 2 = 0 ⇒ 154 a 2 − 154 a + 79 = 0 Sum of values = − 154 ( − 154 ) ​ = 1 ​

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