First slide

Theory of equations

Question

Difference between the corresponding roots of $x^{2} + ax + b = 0 and x^{2} + bx + a = 0$ is same and $a \neq b$ , then

Moderate

A

$a + b + 4 = 0$
B

$a + b - 4 = 0$
C

$a - b - 4 = 0$
D

$a - b + 4 = 0$

Solution

Let $α_{1}, β_{1}$ are the roots of the equation
$x^{2} + ax + b = 0 \Rightarrow x = \frac{- a \pm \sqrt{a^{2} - 4 b}}{2}$
$\Rightarrow α_{1} = \frac{- a + \sqrt{a^{2} - 4 b}}{2}, β_{1} = \frac{- a - \sqrt{a^{2} - 4 b}}{2}$
and $α_{2}, β_{2}$ are the roots of the equation
$x^{2} + bx + a = 0$
So, $α_{2} = \frac{- b + \sqrt{b^{2} - 4 a}}{2}, β_{2} = \frac{- b - \sqrt{b^{2} - 4 a}}{2}$
Now $α_{1} - β_{1} = \sqrt{a^{2} - 4 b}; α_{2} - β_{2} = \sqrt{b^{2} - 4 a}$
Given, $α_{1} - β_{1} = α_{2} - β_{2} \Rightarrow \sqrt{a^{2} - 4 b} = \sqrt{b^{2} - 4 a}$
$\Rightarrow a^{2} - b^{2} = - 4 (a - b) \Rightarrow a + b + 4 = 0$

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