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JEE Advanced2024MathematicsThree Dimensional GeometryActual

A straight line drawn from the point P(1,3,2) , parallel to the line x-2 1 = y-4 2 = z-6 1 , intersects the plane L₁: x-y+3 z=6 at the point Q . Another straight line which passes through Q and is perpendicular to the plane L₁ intersects the plane L₂: 2 x-y+z=-4 at the point R . Then which of the following statements is (are) TRUE?

Options

  1. AThe length of the line segment P Q is 6
  2. BThe coordinates of R are (1,6,3)
  3. CThe centroid of the triangle P Q R is ( 4 3 , 14 3 , 5 3 )
  4. DThe perimeter of the triangle P Q R is 2 + 6 + 11

Correct answer

A. The length of the line segment P Q is 6

Step-by-step solution

aligned & line : x-1 1 = y-3 2 = z-2 1 & (x, y, z)=( +1,2 +3, +2) aligned Put in L₁: x-y+3 z=6 aligned & ( +1)-(2 +3)+3( +2)=6 & 2 =2 =1 & Q=(2,5,3) aligned line : x-2 1 = y-5 -1 = z-3 3 (x, y, z)=(t+2,5-t, 3 t+3) Put in L₂: 2 x-y+z=-4 aligned & 2(t+4)-(5-t)+(3 t+3)=-4 & 6 t=-6 t=-1 & R=(1,6,0) aligned aligned & Perimeter = 6 + 13 + 11 & Centroid = ( 4 3 , 14 3 , 5 3 ) aligned

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