Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced2026MathematicsThree Dimensional GeometryActual

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 point (4, 2, 2) and is parallel to P . Then which of the following statements is (are) TRUE?

Options

  1. AThe equation of the plane P is 7x - 5y + z = -10
  2. BThe distance between the planes P and P₁ is 30
  3. CThe distance of the plane P from the origin is 2 3
  4. DThe acute angle between the plane P and the plane 2x + 2y + z = 3 is ⁻¹ ( 1 3 3 )

Correct answer

A. The equation of the plane P is 7x - 5y + z = -10

Step-by-step solution

Let the normal vector to the plane P be n . Since the plane P contains the line x-1 2 = y-3 3 = z+2 1 , its normal vector n is perpendicular to the line's direction vector b = 2 i + 3 j + k . Since P is perpendicular to the plane x + 2y + 3z = 4 , n is perpendicular to its normal vector n ₁ = i + 2 j + 3 k . n = (2 i + 3 j + k ) ( i + 2 j + 3 k ) = i (9-2) - j (6-1) + k (4-3) = 7 i - 5 j + k Plane P passes through the point (1, 3, -2) which lies on the given line. Equation of plane P is 7(x-1) - 5(y-3) + 1(z+2) = 0

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let 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 = 2026Let a triangle PQR be such that P and Q lie on the line x+3 8 = y-4 2 = z+1 2 and are at a distance of 6 units from R(1, 2, 3) . If ( , , ) is the centroid of PQR , then + + is equ 2026 Full Three Dimensional Geometry list All JEE Advanced PYQs