Multiple choice

Let the positive numbers $a, b, c$ be in $G.P$. and $\omega$ be a complex cube root of unity then the equation $ax^{2}+bx+c=0$ must have

  1. ratio of roots as $1:\omega$
  2. ratio of roots as $b:ac$

  3. real roots

  4. imaginary roots
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A Correct answer
Explanation

For a GP a, b, c, we have b^2 = ac. The roots of ax^2 + bx + c = 0 are x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Substituting b^2 = ac, the discriminant is ac - 4ac = -3ac. The roots are (-b +/- i*sqrt(3ac)) / 2a. The ratio of the roots is indeed related to omega.

AI explanation

Since a, b, and c are in a geometric progression, we can express them as a, ar, and ar^2 respectively, where r is the common ratio. Substituting these into the quadratic equation yields ax^2 + arx + ar^2 = 0, which simplifies to x^2 + rx + r^2 = 0 by dividing by a. Using the quadratic formula, the roots are x = (-r plus or minus sqrt(r^2 - 4r^2)) / 2 = r(-1 plus or minus i*sqrt(3))/2. Factoring out r, the roots become r*omega and r*omega^2 (where omega is a complex cube root of unity), making the ratio of the roots exactly 1:omega.

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