Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202429 Jan 2024Morning ShiftMathematicsThree Dimensional GeometryActual

Let O be the origin and the position vector of A and B be 2 i ^ + 2 j ^ + k ^ and 2 i ^ + 4 j ^ + 4 k ^ respectively. If the internal bisector of ∠ A O B meets the line A B at C , then the length of O C is

Options

  1. A2 3 31
  2. B2 3 34
  3. C3 4 34
  4. D3 2 31

Correct answer

B. 2 3 34

Step-by-step solution

We know that, the internal bisector divides base in the same ratio as the two remaining sides. ⇒ O A O B = A C B C ⇒ 2 2 + 2 2 + 1 2 2 2 + 4 2 + 4 2 = A C B C ⇒ 3 6 = A C B C ⇒ A C : B C = 1 : 2 ⇒ C ≡ 1 × 2 + 2 × 2 3 , 1 × 4 + 2 × 2 3 , 1 × 4 + 2 × 1 3 ⇒ C ≡ 2 , 8 3 , 2 ⇒ O C = 2 2 + 8 3 2 + 2 2 ⇒ O C = 8 + 64 9 = 136 3 ⇒ O C = 2 34 3

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line x-1 2 = y-3 3 = z+2 1 and is perpendicular to the plane x + 2y + 3z = 4 . Let P₁ be the plane which passes through the po 2026Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the foot of the perpendicular drawn from the point R(4, -1, 5) to the line L . Another line pass 2026Let the foot of perpendicular from the point ( , 2, 3) on the line x-4 1 = y-9 2 = z-5 1 be the point (1, , 2) . Then the distance between the lines x-1 2 = y-2 3 = z+4 6 and x- 2 2026Let a line L₁ pass through the origin and be perpendicular to the lines L₂: r = (3+t) i + (2t-1) j + (2t+4) k and L₃: r = (3+2s) i + (3+2s) j + (2+s) k , t, s R . If (a, b, c) , a 2026The shortest distance between the lines x-4 1 = y-3 2 = z-2 -3 and x+2 2 = y-6 4 = z-5 -5 is : 2026Let the image of the point P(0, -5, 0) in the line x-1 2 = y 1 = z+1 -2 be the point R and the image of the point Q (0, -1 2 , 0 ) in the line x-1 -1 = y+9 4 = z+1 1 be the point S 2026Let the image of the point P(1, 6, a) in the line L: x 1 = y-1 2 = z-a+1 b , b > 0 , be ( a 3 , 0, a+c ) . If S( , , ) , > 0 , is the point on L such that the distance of S from th 2026Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 = 2026 Full Three Dimensional Geometry list All JEE Main PYQs