Let α and 2 be the roots of the equation x^2 − px + r = 0 and α/2, 2β be the roots of the equation

Question

Let α and β be the roots of the equation x² − px + r= 0  and α/2, 2β be the roots of the equation x² − qx + r = 0. The value of r is

(a)  2/9 (p - q)(2q - p)

(b)  2/9 (q - p)(2p - q)

(c)  2/9 (q - 2p)(2p - p)

(d)  2/9 (2p - q)(2q - p)

Answer 1

Correct option (d)  2/9 (2p - q)(2q - p)

Since α and β are the roots of equation x² − px + r = 0,

we get

α + β

α + β = p ....(1)

αβ = r  ....(2)

Also, α/2 and 2β since are the roots of equation x² − qx + r = 0, we get

α/2 + 2β = q ⇒ α + 4β = 2q

Solving Eqs. (1) and (2), we get

α = 2(2p - q)/3 and β = 2q - p/3

Substituting α and β in Eq. (2), we get

r = 2/9(2p - q)(2q - p)