Multiple choice

Let $\alpha$ and $\beta$ be the roots of equation $px^2+qx+r=0.p \neq 0.$ If p,q and r in AP and $\cfrac 1 \alpha +\cfrac 1 \beta =4,$ then the value of $|\alpha - \beta|$ is

  1. $\cfrac {\sqrt {61}}{9}$
  2. $\cfrac {2\sqrt 17}{9}$
  3. $\cfrac {\sqrt {34}}{9}$
  4. $\cfrac {2\sqrt 13}{9}$
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D Correct answer
Explanation

p, q, r in AP implies 2q = p + r. Roots alpha, beta satisfy alpha+beta = -q/p and alpha*beta = r/p. 1/alpha + 1/beta = (alpha+beta)/(alpha*beta) = (-q/p) / (r/p) = -q/r = 4. So q = -4r. Since 2q = p + r, -8r = p + r, so p = -9r. Roots are alpha, beta of px^2 + qx + r = 0, which is -9rx^2 - 4rx + r = 0. Divide by -r: 9x^2 + 4x - 1 = 0. Roots = (-4 +/- sqrt(16 - 4*9*(-1))) / 18 = (-4 +/- sqrt(52)) / 18 = (-4 +/- 2*sqrt(13)) / 18 = (-2 +/- sqrt(13)) / 9. Difference = |alpha - beta| = (2*sqrt(13)) / 9.

AI explanation

Since p, q, and r are in arithmetic progression, we have 2q = p + r. The sum of reciprocals of the roots is 1/alpha + 1/beta = (alpha + beta)/(alpha*beta) = (-q/p)/(r/p) = -q/r = 4, which implies r = -q/4. Substituting this into the arithmetic progression relation gives 2q = p - q/4, or p = 9q/4. Using the identity (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta, we get (-q/p)^2 - 4(r/p), which simplifies to (-4/9)^2 - 4(-1/9) = 16/81 + 4/9 = 52/81. Taking the square root gives the absolute difference as 2*sqrt(13)/9.

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