Multiple choice

lf $\alpha,\ \beta$ are real and $\alpha^{2},-\beta^{2}$ are the roots of the equation $\mathrm{a}^{2}\mathrm{x}^{2}+\mathrm{x}+(1-\mathrm{a}^{2})=0(\mathrm{a}>1)$, then $\beta^{2}=$

  1. $ \mathrm{a}^{2}$
  2. $1$
  3. $1-\mathrm{a}^{2}$
  4. $1+\mathrm{a}^{2}$
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B Correct answer
Explanation

For the equation ax^2 + x + (1-a^2) = 0, the roots are alpha^2 and -beta^2. By Vieta's formulas, the product of roots is (1-a^2)/a^2. Thus, (alpha^2)(-beta^2) = (1-a^2)/a^2. Also, the sum of roots is -1/a^2. Given the structure, testing beta^2 = 1 yields roots that satisfy the equation.

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