Answer :

Three points are said to be collinear if they all lie in a straight line. 
If Three points (x1y1), (x2y2), (x3y3) are collinear then no triangle can be formed using three points and so the area formed by the triangle by the three points is zero.


Area of Triangle = 1/2 [x(y− y3x(y− y1x(y− y2)]                           ...(1)

1. For triangle, A(-1, -1), B(0, 1), C(1, 3)
Area =
 
       = 1/2 [ -1(-2) + 0(3+1) + 1(-1-1)]
      
      = 1/2 [2 + 0 - 2]
  
     = 1/2 [2-2]
     = 0
Hence the points are collinear

2. For triangle, D(-2, -3), E(1, 0), F(2, 1)
Using 1,

Area = 1/2 [-2(0-1) + 1(1-(-3)) + 2(-3-0)]
       = 1/2 [-2(-1) + 1(1+3) + 2(-3)]
      = 1/2 [ 2 + 4 -6 ]
     = 1/2 [ 6 - 6 ]
     = 1/2 (0)
    = 0
Hence the points are collinear.


3. For triangle, L(2, 5), M(3, 3), N(5, 1)
Using 1,

Area = 1/2 [2(3-1) + 3(1-5) + 5(5-3)]
       = 1/2 [ 2(2) + 3(-4) + 5(2)]
      = 1/2 [ 4 - 12 + 10 ]
     = 1/2 [ 14 - 12 ]
    
     = 1/2 (2)
    = 1

Hence the points are not collinear.

4. For triangle, P(2, -5), Q(1, -3), R(-2, 3)
Using 1,
Area =
= 0
Area = 1/2 [ 2(-3-3) + 1(3-(-5)) + (-2)(-5-(-3))]
       = 1/2 [ 2(-6) + 1(3+5) - 2 (-5+3)]
      = 1/2 [ -12 + 8 -2(-2)]
     = 1/2 [-12 + 8 +4]
    = 1/2 [ -12 + 12]
   = 1/2 (0)

   = 0

Hence the points are collinear.


5. For triangle, R(1, -4), S(-2, 2), T(-3, 4)
Area =


Hence the points are collinear.


6. For triangle, A(-4, 4), K(-2, 5/ 2 ), N(4, -2)
Area =


Hence the points are collinear.

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