Question

If \(z_1\) and \(z_2\) be two roots of the equation \(z^2 + az + b = 0, \, a^2 < 4b\), then the origin, \(z_1\) and \(z_2\) form an equilateral triangle if:

• A. \(a^2 = 3b^2\)
• B. \(a^2 = 3b\)
• C. \(b^2 = 3a\)
• D. \(a^2 = b^2\)

Hint:

When roots form geometric shapes such as equilateral triangles, use geometric properties and the relation between the coefficients of the quadratic equation to derive the required conditions.

Answer 1

The Correct Option is B

Step 1: We are given that \( z_1 \) and \( z_2 \) are the roots of the quadratic equation:

\[ z^2 + az + b = 0 \]

From the quadratic formula, the roots \( z_1 \) and \( z_2 \) are:

\[ z_1, z_2 = \frac{-a \pm \sqrt{a^2 - 4b}}{2} \]

Step 2: Since \( a^2 < 4b \), the discriminant is negative, implying that the roots are complex.

Now, for the points \( z_1 \), \( z_2 \), and the origin to form an equilateral triangle, the condition is that the angle between the vectors \( \overrightarrow{0z_1} \) and \( \overrightarrow{0z_2} \) should be \( 60^\circ \).

Step 3: The geometric condition for forming an equilateral triangle is that the distance between the origin and each of the roots \( z_1 \) and \( z_2 \) should be equal, and the angle between the vectors should be \( 60^\circ \). This condition leads to the relation:

\[ a^2 = 3b \]

Step 4: Therefore, the correct relation between \( a \) and \( b \) is \( a^2 = 3b \).