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Graph of Quadratic Expression

If α, β are...

Question

If $α, β$ are the quadratic equation $x^{2} + a x + b = 0, (b \neq 0)$ ; then the quadratic equation whose roots are $α - \frac{1}{β}, β - \frac{1}{α}$ is

A

a x^{2} + a (b - 1) x + (a - 1)^{2} = 0

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B

b x^{2} + a (b - 1) x + (b - 1)^{2} = 0

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C

x^{2} + a x + b = 0

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D

a b x^{2} + b x + a = 0

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Solution

The correct option is B $b x^{2} + a (b - 1) x + (b - 1)^{2} = 0$
roots are $α & β$
sum = $α + β$ = -a
product = $α β$ = b
new roots are $α - \frac{1}{β}, β - \frac{1}{α}$
sum of roots =
$= α - \frac{1}{β} + β - \frac{1}{α} = α + β - (\frac{1}{α} + \frac{1}{β}) = (α + β) - (\frac{α + β}{α β}) = - a + \frac{a}{b} = \frac{a - a b}{b}$
product of roots =
$(α - \frac{1}{β}) * (β - \frac{1}{α}) = α β - 1 - 1 + \frac{1}{α β} = b + \frac{1}{b} - 2 = \frac{b^{2} - 2 b + 1}{b}$

Equation:
$x^{2}$ -(sum of roots)x+product of roots=0

$x^{2} - (\frac{a - a b}{b}) x + \frac{b^{2} - 2 b + 1}{b} = 0$

${b x}^{2} + a (b - 1) x + {(b - 1)}^{2} = 0$

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