First slide

Theory of equations

Question

If $α, β$ be the roots of $x^{2} + px + q = 0 and α + h, β + h$ are the roots of $x^{2} + rx + s = 0,$ then

Moderate

A

$\frac{p}{r} = \frac{q}{s}$
B

$2 h = [\frac{p}{q} + \frac{r}{s}]$
C

$p^{2} - 4 q = r^{2} - 4 s$
D

${pr}^{2} = {qs}^{2}$

Solution

$α + β = - p, αβ = q$
$α + β + 2 h = - r, (α + h) (β + h) = s$
$- p + 2 h = - r \Rightarrow h = \frac{p - r}{2} . . . (i)$
Now, $αβ + h (α + β) + h^{2} = s$
$\Rightarrow q + h (- p) + h^{2} = s$
$\Rightarrow q + (\frac{p - r}{2}) (- p) + {(\frac{p - r}{2})}^{2} = s$
$\Rightarrow q - \frac{(p^{2} - pr)}{2} + \frac{p^{2} + r^{2} - 2 pr}{4} = s$
$\Rightarrow 4 q - 2 p^{2} + 2 pr + p^{2} + r^{2} - 2 pr = 4 s$
$\Rightarrow 4 q - p^{2} + r^{2} - 4 s = 0 \Rightarrow r^{2} - 4 s = p^{2} - 4 q .$

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