Question

If \(p\) and \(q\) are non-zero real numbers and \(\alpha^3 + \beta^3 = -p,\ \alpha\beta = q\), then a quadratic equation whose roots are \(\frac{\alpha^2}{\beta},\ \frac{\beta^2}{\alpha}\) is :

• A. \(px^2 - qx + p^2 = 0\)
• B. \(qx^2 + px + q^2 = 0\)
• C. \(px^2 + qx + p^2 = 0\)
• D. \(qx^2 - px + q^2 = 0\)

Hint:

Find the sum and product of the new roots in terms of \(\alpha^3+\beta^3\) and \(\alpha\beta\).

Answer 1

The Correct Option is B

Sum of new roots \(= \frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha} = \frac{\alpha^3+\beta^3}{\alpha\beta} = \frac{-p}{q}\).
Product \(= \frac{\alpha^2}{\beta}\cdot\frac{\beta^2}{\alpha} = \alpha\beta = q\).
Equation: \(x^2 - (\text{sum})x + \text{product} = 0 \Rightarrow x^2 + \frac{p}{q}x + q = 0\).
Multiplying by \(q\): \(qx^2 + px + q^2 = 0\).
Answer: (2) \(qx^2 + px + q^2 = 0\)