If - and - are the roots of a x2-b x-c-0b-0 and -0- Then the roots of -x-2-x-2-0 are

The correct option is A real and of opposite signb≠0⇒α+β≠0 For α(x β)^2+β(x α)^2=0 ⇒ (α+β)x^2 4αβ x+αβ(α+β)=0 Δ=16α^2β^2 4αβ(α+β)^2 gt;0 (∵αβ lt;0) produ ...

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Standard XII

Mathematics

Range

If α and β ar...

Question

If $α$ and $β$ are the roots of $a x^{2} + b x + c = 0 (b \neq 0)$ and $α β < 0.$ Then the roots of $α (x - β)^{2} + β (x - α)^{2} = 0$ are

real and of opposite sign

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negative

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positive

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non-real

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Solution

The correct option is A real and of opposite sign
$b \neq 0 \Rightarrow α + β \neq 0$
For
$α (x - β)^{2} + β (x - α)^{2} = 0$
$\Rightarrow (α + β) x^{2} - 4 α β x + α β (α + β) = 0$

$Δ = 16 α^{2} β^{2} - 4 α β (α + β)^{2} > 0 (∵ α β < 0)$

$product of roots = \frac{α β (α + β)}{(α + β)} = α β < 0$

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Similar questions

Q. If

α, β

are roots of equation

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

which are real and opposite in sign, then roots of the equation

α {(x - β)}^{2} + β {(x - α)}^{2} = 0

are

α, β

are the roots of

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

L_{1} = L t x \to α \frac{1 - c o s ({a x}^{2} + b x + c)}{{(x - α)}^{2}}

L_{2} = L t x \to β \frac{1 - c o s ({a x}^{2} + b x + c)}{{(x - β)}^{2}}

L_{3} = L t (x - α) (x - β) \to 0 \frac{1 - c o s ({a x}^{2} + b x + c)}{{(x - α)}^{2} {(x - β)}^{2}}

Q. If

α

and

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are roots of

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

then the equation whose roots are

α^{2}

and

β^{2}

Q. If

α, β

are the roots of the equation

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

, show that

log (a - b x + c x^{2}) = log a + (α + β) x - \frac{α^{2} + β^{2}}{2} x^{2} + \frac{α^{3} + β^{3}}{3} x^{3} - . . .

Q. If

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If α and β are the roots of ax2+bx+c=0 (b≠0) and αβ<0. Then the roots of α(x−β)2+β(x−α)2=0 are

The correct option is A real and of opposite signb≠0⇒α+β≠0 For α(x−β)2+β(x−α)2=0 ⇒(α+β)x2−4αβx+αβ(α+β)=0 Δ=16α2β2−4αβ(α+β)2>0 (∵αβ<0) product of roots=αβ(α+β)(α+β)=αβ<0

If $α$ and $β$ are the roots of $a x^{2} + b x + c = 0 (b \neq 0)$ and $α β < 0.$ Then the roots of $α (x - β)^{2} + β (x - α)^{2} = 0$ are