First slide

Theory of equations

Question

Let $a, β$ be the roots of the equation $x^{2} - p x + r = 0$ and $\frac{α}{2}, 2 β$ be the roots of the equation $x^{2} - q x + r$ $= 0 .$ Then the value of $r$ is :

Moderate

A

$\frac{2}{9} (p - q) (2 q - p)$
B

$\frac{2}{9} (q - p) (2 p - q)$
C

$\frac{2}{9} (q - 2 p) (2 q - p)$
D

$\frac{2}{9} (2 p - q) (2 q - p)$

Solution

$α + β = p, α β = r$
and    $\frac{α}{2} + 2 β = q$
But    $α + β = p and α + 4 β = 2 q$
$\Rightarrow β = \frac{1}{3} (2 q - p) and α = \frac{2}{3} (2 p - q)$
Thus,    $a β = r$
$\Rightarrow \frac{2}{9} (2 p - q) (2 q - p) = r$

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