Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsThree Dimensional Geometry

Let the projections of a point P on the x -axis and y -axis be A (7,0,0) and B (0,1,0) respectively. If the foot of the perpendicular drawn from P to the line L ₁: x-1 2 = y+1 1 = z-2 3 is Q (3,0,5) , then the square of the perpendicular distance from P to the line L ₂: x-2 1 = y -1 = z-1 2 is equal to :

Options

  1. A27
  2. B21
  3. C26
  4. D45

Correct answer

B. 21

Step-by-step solution

Since the projections of P on the x -axis and y -axis are (7,0,0) and (0,1,0) respectively, the coordinates of P can be taken as (7,1,c) . The foot of the perpendicular from P to L ₁ is given as Q (3,0,5) . The direction ratios of the line L ₁ are 2, 1, 3 . The vector PQ is (3-7) i + (0-1) j + (5-c) k = -4 i - j + (5-c) k . Since PQ is perpendicular to L ₁ , their dot product must be zero: 2(-4) + 1(-1) + 3(5-c) = 0 -8 - 1 + 15 - 3c = 0 6 - 3c = 0 c = 2 Thus, the point P is (7,1,2) . Now, we need to find the square

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