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Consider the following statements in respect of the roots of the equation x3 - 8 = 0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct?

Consider the following statements in respect of the roots of the equation x3 - 8 = 0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct? Correct Answer 1 only

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.

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