Multiple choice

If roots of cubic equation are in G.P. , $ ax^3 + bx^2 + cx + d $ then :

  1. ${c^3}a = {b^3}d$
  2. $a^2 c = b^2 d $
  3. $ ac^2 = bd^2 $

  4. NOTA
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A Correct answer
Explanation

If roots are in GP, let them be k/r, k, kr. The product of roots is -d/a = k^3. The sum of roots taken two at a time is c/a = k^2/r + k^2 + k^2r = k^2(1/r + 1 + r). The sum of roots is -b/a = k(1/r + 1 + r). Thus, c/a = (k)(-b/a), so c/a = -b/a * k. Since k = (-d/a)^(1/3), this leads to the relation c^3 * a = b^3 * d.

AI explanation

Let the roots of the cubic equation in geometric progression be k/r, k, and kr. Using Vieta's formulas, the product of the roots gives k^3 = -d/a, the sum of the roots gives k(r + 1 + 1/r) = -b/a, and the sum of the roots taken two at a time gives k^2(r + 1/r + 1) = c/a. Multiplying the sum of roots by k yields k^2(r + 1/r + 1) = -bk/a, and equating this to the third equation results in c/a = -bk/a, which simplifies to k = -c/b. Substituting k = -c/b into k^3 = -d/a gives (-c/b)^3 = -d/a, resulting in a*c^3 = b^3*d.

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