Multiple choice

If the ratio of the roots of the equation $p{ x }^{ 2 }+qx+r=0\quad $ is $a: b$, then $\cfrac { ab }{ { \left( a+b \right) }^{ 2 } } $ is equal to

  1. $\cfrac { { p }^{ 2 } }{ qr } $
  2. $\cfrac { pr }{ { q }^{ 2 } } $
  3. $\cfrac { { q }^{ 2 } }{ pr } $
  4. $\cfrac { pq }{ { r }^{ 2 } } $
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B Correct answer
Explanation

For roots r1, r2 with ratio a:b, r1/r2 = a/b. Sum = -q/p, Product = r/p. (r1+r2)^2 / (r1*r2) = (-q/p)^2 / (r/p) = q^2 / pr. Also (r1+r2)^2 / (r1*r2) = (a+b)^2 / ab. Thus ab/(a+b)^2 = pr/q^2.

AI explanation

Using the sum and product of roots formulas for px^2 + qx + r = 0, the sum of the roots (ak + bk) is -q/p and the product of the roots (ak times bk) is r/p. The target expression ab / (a + b)^2 can be rewritten by multiplying by k^2 on both top and bottom, yielding abk^2 / (ak^2 + 2abk + b^2k^2). The numerator becomes the product of the roots, r/p. The denominator becomes the square of the sum of the roots, q^2/p^2. Dividing the numerator by the denominator gives (r/p) / (q^2/p^2), which simplifies to pr/q^2.

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