Multiple choice

Let z1 and z2 be the roots of the equation z2 + pz + q = 0, where p and q are real. The points represented by z1, z2 and the origin form an equilateral triangle, if

  1. p2 = 3q

  2. p2 > 3q

  3. p2 < 3q

  4. p2 = 2q

  5. p = 3q
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For roots z1, z2 and the origin to form an equilateral triangle, the condition is z1^2 + z2^2 - z1*z2 = 0. Since z1+z2 = -p and z1*z2 = q, we substitute: (z1+z2)^2 - 3*z1*z2 = 0, which gives p^2 - 3q = 0, or p^2 = 3q.

AI explanation

For the origin and the two complex roots to form an equilateral triangle, the area of the triangle must be proportional to the square of the distance between the roots. The distance between the roots is the square root of the discriminant, which is sqrt(p^2 - 4q). The magnitude of the roots is sqrt(q), and the area of the equilateral triangle formed on the coordinate plane dictates that p^2 - 3q = 0. Rearranging this geometric condition yields the required algebraic relationship p^2 = 3q.

Keep practicing — related questions