Multiple choice

Find the condition to be satisfied by the coefficients of the equation $px^2+qx+r=0$, so that the roots are in the ratio 3:4.

  1. $12q^2=49pr$
  2. $12q^2=-49pr$
  3. $49q^2=12pr$
  4. $49q^2=-12pr$
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A Correct answer
Explanation

If roots are in ratio 3:4, let them be 3k and 4k. Sum of roots = 7k = -q/p. Product of roots = 12k^2 = r/p. From the sum, k = -q/(7p). Substitute into the product: 12 * (-q/7p)^2 = r/p. 12 * (q^2 / 49p^2) = r/p. 12q^2 = 49pr.

AI explanation

Assume the roots of the quadratic equation are 3m and 4m based on the 3:4 ratio. The sum of the roots gives 3m + 4m = -q/p, which simplifies to 7m = -q/p, so m = -q/(7p). The product of the roots gives (3m) * (4m) = r/p, which simplifies to 12m^2 = r/p. Substitute m^2 = q^2/(49p^2) into the product equation to get 12(q^2/(49p^2)) = r/p. Multiplying both sides by 49p^2 yields the condition 12q^2 = 49pr.

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