Question: If α, β be the roots of x<sup>2</sup> + px – q = 0 &γ, δ be the roots of x<sup>2</sup> + px + r = ...

Question

If α, β be the roots of x² + px – q = 0 &γ, δ be the roots of

x² + px + r = 0 then $\frac{(\alpha - \gamma)(\alpha - \delta)}{(\beta - \gamma)(\beta - \delta)}$ =

Answer 1

$\begin{array} { l } \alpha + \beta = - p \\ \gamma + \delta = - p \end{array} \quad \Rightarrow \alpha + \beta = \gamma + \delta$

Now, (α − γ) (α − δ) = α² − α (γ + δ) + γδ

= α² − α (α + β) + r

= −αβ + r = q + r

By symmetry of result,

(β – γ) (β – δ) = q + r

Hence ratio is 1