First slide

Theory of equations

Question

If $a (p + q)^{2} + 2 bpq + c = 0$ and $a (p + r)^{2} + 2 bpr + c = 0$ $(a \neq 0)$ , then

Moderate

A

$qr = p^{2}$
B

$qr = p^{2} + \frac{c}{a}$
C

$qr = - p^{2}$
D

none of these

Solution

Given, $a (p + q)^{2} + 2 bpq + c = 0 and a (p + r)^{2} + 2 bpr + c = 0$
$\Rightarrow q and r satisfy the equation a (p + x)^{2} + 2 bpx + c = 0$
$\Rightarrow q and r are the roots of {ax}^{2} + 2 (ap + bp) x + c + {ap}^{2} = 0$
$\Rightarrow qr = Product of roots = \frac{c + {ap}^{2}}{a} = p^{2} + \frac{c}{a}$

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