Concept: Non-Collinear Points: The set of points that do not lie on the same line are called non-collinear points. We cannot draw a single straight line through these points. Formula Used: a3 - b3 = (a - b) (a2 + ab + b2) Calculation: Given equation x3 - 8 = 0 ⇒ x3 - 23 = 0 ⇒ (x - 2) (x2 + 2x + 4) = 0 ⇒ x - 2 = 0 or x2 + 2x + 4 = 0 ⇒ x = 2 or x = -1 + i√3 or x = -1 - i√3 We can take these three points on complex plane So, we can write 1st point as 2 + i.0, -1 + i√3, -1 - i√3 The three points on complex plain is (2, 0), (-1,√3), (-1, -√3) Case I. Equation of line passes through (-1,√3), (-1, -√3) is (y - √3) = (2√3/0) (x + 1) ⇒ x + 1 = 0 And this line does not passes through (2, 0). ∴ These points (roots) are not collinear. Case 2. We know that these points are non-collinear it means they will definitely pass through a circle Let the equation of circle be (x - a)2 + (y - b)2 = r2 Now, According to our assumption (2 - a)2 + (0 - b)2 = r2 ----(i) (-1 - a)2 + (√3 - b)2 = r2 ----(ii) (-1 - a)2 + (-√3 - b)2 = r2 ----(iii) On solving (i), (ii) and (iii), we get a = 0, b = 0 and r = 2 Hence the equation of circle is (x - 0)2 + (y - 0)2 = 22 ∴ The roots represents a circle but radius of circle is 2.