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

Let a line L pass through the point A(1, 1, 1) and be perpendicular to the lines L₁ : x-4 1 = y 2 = z-2 -1 and L₂ : x 2 = y-1 1 = z+1 1 . If P is the point of intersection of the line L and the plane : x + y + z = 6 , and Q is the foot of the perpendicular drawn from P onto the line L₁ , then the value of PQ^2 is equal to ________.

Correct answer

56

Step-by-step solution

The direction ratios of the line L are proportional to the cross product of the direction vectors of L₁ and L₂ . Let d₁ = i + 2 j - k and d₂ = 2 i + j + k . d = d₁ d₂ = vmatrix i & j & k 1 & 2 & -1 2 & 1 & 1 vmatrix = 3 i - 3 j - 3 k So, the direction ratios of L can be taken as (1, -1, -1) . The equation of line L passing through A(1, 1, 1) is: x-1 1 = y-1 -1 = z-1 -1 = Any point on L is (1+ , 1- , 1- ) . Since this point P lies on the plane x + y + z = 6 , we have: (1+ ) + (1- ) + (1- ) = 6 3 - = 6 = -3 Thus, the

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