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JEE Main202528 Jan 2025Morning ShiftMathematicsThree Dimensional GeometryActual

Let A (x, y, z) be a point in x y -plane, which is equidistant from three points (0,3,2),(2,0,3) and ( 0,0,1 ). Let B =(1,4,-1) and C =(2,0,-2) . Then among the statements (S1) : ABC is an isosceles right angled triangle, and (S2) : the area of ABC is 9 2 2 ,

Options

  1. Aboth are true
  2. Bonly (S2) is true
  3. Conly (S1) is true
  4. Dboth are false

Correct answer

C. only (S1) is true

Step-by-step solution

aligned & A ( x , y , z ) Let P (0,3,2), Q (2,0,3), R (0,0,1) & AP = AQ = AR & x ^2+( y -3)^2+( z -2)^2=( x -2)^2+ y ^2+( z -3)^2= x ^2+ & y ^2+( z -1)^2 aligned In x y plane z=0 So, x^2-4 x+4+y^2+9=x^2+y^2+1 aligned & x=3 & 9+y^2-6 y+9+4=x^2+y^2+1 aligned So, A (3,2,0) also B (1,4,-1) & C (2,0,-2) Now A B= 4+4+1 =3 aligned & AC = 1+4+4 =3 & BC = 1+16+1 = 18 aligned AB = AC isosceles & AB ^2+ AC ^2= BC ^2 right angle Area of ABC = 1 2 base.height 1 2 3 3= 9 2 So only S₁ is true

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