First slide

Theory of equations

Question

If $α, β$ are the roots of $a x^{2} + b x + c = 0$ and $α + h, β + h$ are the roots of $p x^{2} + q x + r = 0,$ then $h =$

Moderate

A

$(\frac{b}{a} - \frac{q}{p})$
B

$\frac{1}{2} (\frac{b}{a} - \frac{q}{p})$
C

$- \frac{1}{2} (\frac{b}{a} - \frac{q}{p})$
D

none of these

Solution

We have, $α + β = - \frac{b}{a}, α β = \frac{c}{a}$
and $α + h + β + h = - \frac{q}{p^{2}}, (α + h) (β + h) = \frac{r}{p}$
$\Rightarrow α + β + 2 h = - \frac{q}{p} \Rightarrow - \frac{b}{a} + 2 h = - \frac{q}{p} [∵ α + β = - \frac{b}{a}]$
$\Rightarrow h = \frac{1}{2} (\frac{b}{a} - \frac{q}{p})$

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