Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202329 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

The plane 2 x-y+z=4 intersects the line segment joining the points A ( a ,-2,4) and B (2, ~b ,-3) at the point C in the ratio 2: 1 and the distance of the point C from the origin is 5 . If a b<0 and P is the point ( a - b , b , 2 ~b - a ) then CP ^2 is equal to:

Options

  1. A17 3
  2. B16 3
  3. C73 3
  4. D97 3

Correct answer

A. 17 3

Step-by-step solution

Given points are: A ( a , - 2 , 4 ) and B ( 2 , &#160; b , - 3 ) Point C divides the line segment joining points A and B in the ratio 2 : 1 , so C &#8801; a + 4 3 , 2 b - 2 3 , - 2 3 Also, C lies on 2 x - y + 2 = 4 , therefore 2 a + 8 3 - 2 b - 2 3 - 2 3 = 4 &#8658; 2 a + 8 - 2 b + 2 - 2 = 12 &#8658; a - b = 2 &#160; &#160; &#160; . . . . 1 Also, O C = 5 &#8658; a + 4 3 2 + 2 b - 2 3 2 + 4 9 = 5 &#8658; a 2 + 8 a + 16 + 4 b 2 + 4 - 8 b = 45 &#8658; a 2 + 8 a + 4 b 2 - 8 b = 25 &#160; &#160; &#160; . . . 2 Solving 1

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