Multiple choice

If $\alpha$ and $\beta$ are the roots of $ax^2+c=bx$, then the equation $(a+cy)^2=b^2y$ in $y$ has the roots

  1. $\alpha ^{-1}, \beta ^{-1}$
  2. $\alpha ^{2}, \beta ^{2}$
  3. $\alpha \beta^{-1}, \alpha ^{-1}\beta$
  4. $\alpha ^{-2}, \beta ^{-2}$
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D Correct answer
Explanation

The roots of ax^2 - bx + c = 0 are alpha and beta. The equation (a + cy)^2 = b^2y can be rewritten by substituting y = 1/x^2. The roots of the transformed equation are the squares of the reciprocals of the original roots, which is alpha^-2 and beta^-2.

AI explanation

Rewriting the first equation as ax^2 - bx + c = 0, we know the sum of the roots is a+b = b/a and the product is ab = c/a. The equation in y can be rewritten as a^2 + 2acy + c^2y^2 = b^2y, which becomes c^2y^2 + (2ac - b^2)y + a^2 = 0. The product of the roots in this y equation is a^2/c^2. Using the relationship a^2/c^2 = (ab)^2/(ab)^2, we can express the roots as 1/a^2 and 1/b^2, which are a^-2 and b^-2.

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