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Sum and Product of Roots of a Quadratic Equation

If α and ...

Question

If $α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$ and if $p x^{2} + q x + r = 0$ has roots $\frac{1 - α}{α}$ and $\frac{1 - β}{β}$ , then $r$ is

a + 2 b

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a + b + c

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a b + b c + c a

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a b c

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Solution

The correct option is B $a + b + c$
The equation with roots $\frac{1}{α}$ and $\frac{1}{β}$
$= a {(\frac{1}{x})}^{2} + b (\frac{1}{x}) + c$
$= c x^{2} + b x + a = 0$ ..(1)
Now $\frac{1 - α}{α} = \frac{1}{α} - 1$
Similarly $\frac{1 - β}{β} = \frac{1}{β} - 1$
Therefore the quadratic equation containing these roots is
$c {(x + 1)}^{2} + b (x + 1) + a$
$= c x^{2} + (b + 2 c) x + a + b + c = 0$
By comparing coefficients we get with $p x^{2} + q x + r = 0$ we get
$r = a + b + c$

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