Quad equation question pls tell fast - Maths - Complex Numbers and Quadratic Equations

Question

Quad equation question pls tell fast

Ankush Jain · Accepted Answer

Let the roots of given equation $x^{2} - p x + q = 0$ be α and β.

⇒ Sum of roots = α + β = p ......(1)

and product of roots = α β = q .......(2)

According to the given condition,

$α + β = m (α - β) \Rightarrow p = m (α - β) [using (1)] \Rightarrow α - β = \frac{p}{m} ...... (3)$

Adding (1) and (3), we get

$2 α = p + \frac{p}{m} = \frac{p m + p}{m} \Rightarrow α = \frac{p m + p}{2 m}$

On putting value of α in (3), we get

$\frac{p m + p}{2 m} - β = \frac{p}{m} \Rightarrow \frac{p m + p}{2 m} \frac{- p}{m} = β \Rightarrow β = \frac{p m + p - 2 p}{2 m} = \frac{p m - p}{2 m}$

On putting value of α and β in (2), we get

$(\frac{p m + p}{2 m}) (\frac{p m - p}{2 m}) = q \frac{{(p m)}^{2} - p^{2}}{{(2 m)}^{2}} = q \Rightarrow p^{2} (m^{2} - 1) = 4 m^{2} q$

Hence Proved

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