Question

If \( \alpha \) is the real root and \( \beta \), \( \gamma \) are the other roots of the equation \( x^3 - a^3 = 0 \, (a>0) \), then the number of common points of the curves given by \( |z - \beta| = \frac{\sqrt{3a}}{2} \) and \( |z - \gamma| = \frac{\sqrt{3a}}{2} \) is:

• A. 0
• B. 2
• C. 3
• D. 1

Hint:

To find the number of common points of two circles in the complex plane, check the distance between their centers and compare it to the sum and difference of their radii.

Answer 1

The Correct Option is D

We are given that the roots of the equation \( x^3 - a^3 = 0 \) are \( \alpha = a \), \( \beta = -\frac{a}{2} + i\frac{\sqrt{3}}{2}a \), and \( \gamma = -\frac{a}{2} - i\frac{\sqrt{3}}{2}a \). These roots are located on the complex plane. The curves \( |z - \beta| = \frac{\sqrt{3a}}{2} \) and \( |z - \gamma| = \frac{\sqrt{3a}}{2} \) represent circles centered at \( \beta \) and \( \gamma \), respectively, with radius \( \frac{\sqrt{3a}}{2} \). The distance between \( \beta \) and \( \gamma \) is: \[ \text{distance} = \left| \beta - \gamma \right| = \left| -\frac{a}{2} + i\frac{\sqrt{3}}{2}a - \left( -\frac{a}{2} - i\frac{\sqrt{3}}{2}a \right) \right| = a \] Thus, the two circles intersect at exactly one point, since their radius is half the distance between the centers. Therefore, the number of common points is 1. Thus, the correct answer is 1.