Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
MHT CET202525 Apr 2025Evening ShiftMathematicsThree Dimensional GeometryActual

The Cartesian equation of the plane r =(2 i -3 j )+ ( i +2 j - k )+ (2 i +3 j + k ) is

Options

  1. A5 x-4 y+z=22
  2. B5 x-3 y+z=19
  3. C5 x-3 y-z=19
  4. D5 x-4 y-z=22

Correct answer

C. 5 x-3 y-z=19

Step-by-step solution

The plane is defined by the vector equation r = (2 i - 3 j ) + ( i + 2 j - k ) + (2 i + 3 j + k ) , indicating it passes through the point (2, -3, 0) and is parallel to vectors (1, 2, -1) and (2, 3, 1) . The normal vector is found using the cross product: n = (1, 2, -1) (2, 3, 1) = 5 i - 3 j - k . Using the point-normal form with (x₀, y₀, z₀) = (2, -3, 0) and normal (5, -3, -1) , the Cartesian equation is 5(x - 2) - 3(y + 3) - z = 0 , which simplifies to 5x - 3y - z = 19 . This matches option C .

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

The circum radius of the triangle formed by the points (2,-1,1),(1,-3,-5) and (3,-4,-4) is 2025Let A (2,3,5), B (-1,3,2), C ( , 5, ) be the vertices of ABC . If the median through the vertex A is equally inclined to the coordinate axes, then 2025Equation of the plane passing through the origin and perpendicular to the planes x+2 y-z=1 and 3 x-4 y+z=5 is 2025If the line joining the points i +2 j and j -2 k intersects the plane passing through the points 2 i - j , 2 j +3 k and k -2 i at r , then r ( i + j + k )= 2025The vector equation of a plane passing through the line of intersection of the planes r ( i -2 k )=3, r (2 j + k )=5 and the point i +2 j +3 k is 2025The points A (-1,2,3), B (2,-3,1), C (3,1,-2) 2025The direction cosines of the line making angles 4 , 3 and (0 < < 2 ) respectively with X , Y and Z axes are 2025If the equation of the plane passing through the point (3,2,5) and perpendicular to the planes 2 x-3 y+5 z=7 and 5 x+2 y-3 z=11 is x+b y+c z+d=0 , then 2 b+3 c+d= 2025 Full Three Dimensional Geometry list All MHT CET PYQs