Multiple choice

\begin{array} { l } { \text { If } \alpha , \beta \text { are the roots of } a x ^ { 2 } + b x + c , \alpha + h } { \beta + h \text { are the roots of } p x ^ { 2 } + q x + r = 0 } \{ \text { and } D _ { 1 } \text { , } D _ { 2 } \text { are the respective } } \ { \text { discriminants of these equations, } } \ { \text { then } D _ { 1 } : D _ { 2 } = } \end{array}

  1. $\cfrac{a ^ { 2 } }{ p ^ { 2 }}$
  2. $\cfrac{b ^ { 2 }} { q ^ { 2 }}$
  3. $\cfrac{c ^ { 2 } }{r ^ { 2 }}$
  4. $1$
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A Correct answer
Explanation

If roots are shifted by h, the new equation is a(x-h)^2 + b(x-h) + c = 0. The discriminant of a(x-h)^2 + b(x-h) + c = 0 is the same as ax^2 + bx + c = 0 because shifting does not change the gap between roots. However, the coefficients p, q, r are scaled. The ratio of discriminants D1/D2 = (a/p)^2.

AI explanation

The discriminant of the first equation is D1 = b^2 - 4ac. If the roots of the second equation are obtained by adding a constant h to the roots of the first, its discriminant remains unchanged, meaning D2 equals b^2 - 4pr. By Vieta's formulas for the shifted roots, (alpha + h) + (beta + h) = -q/p, which simplifies to (alpha + beta) + 2h = -q/p, so -b/a + 2h = -q/p. Solving for the shift gives 2h = (bq - pa) / (ap), and substituting this back into the shifted product formula reveals that D1/D2 equals a^2 / p^2.

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