If $a, b$ are roots of the equation $x^{2} + p x - q = 0$ and $c, d$ are roots of $x^{2} + p x + r = 0$ , then the value of $(a - c) (a - d)$ is-

q - r

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q + r

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p + r

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p - r

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Solution

The correct option is B $q + r$
Given that, $a$ and $b$ are roots of the equation $x^{2} + p x - q = 0$
So, $a + b = - p$ .....(i)
$a b = - q$ .....(ii)
Also, given that $c$ and $d$ are roots of $x^{2} + p x + r = 0$
So, $c + d = - p$ .....(iii)
$c d = r$ .....(iv)
To find : value of $(a - c) (a - d)$
Solution : $(a - c) (a - d) = a^{2} - a d - a c + c d$
$= a^{2} - a (c + d) + c d$
From equation (i) and (iii),it is clear that $a + b = c + d$
So, $(a - c) (a - d) = a^{2} - a (a + b) + c d$
$= a^{2} - a^{2} - a b + c d$
$= c d - a b$
Putting values of $a b$ and $c d$ , we get $(a - c) (a - d) = r + q = q + r$

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