If - and - are roots of x2 - ax - b - 0 and - - arethe roots of x2 - ax - b - 0- then - - - -

Α+β= a , αβ= b , γ+δ= a nbsp; γδ=b therefore (α γ) (β δ) (α δ) (β γ) nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nb ...

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Byju's Answer

Standard XIII

Mathematics

Relations between Roots and Coefficients

If ∝ and β ar...

Question

If ∝ and β are roots of $x^{2}$ + ax - b = 0 and γ, δ arethe roots of $x^{2}$ + ax + b = 0, then (α-γ) (β-δ) (α-δ) (β-γ)

4 $b^{2}$

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2 $b^{2}$

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Solution

The correct option is A
4 $b^{2}$

α+β= -a , αβ= -b , γ+δ=-a γδ=b

therefore (α-γ) (β-δ) (α-δ) (β-γ)

=[ αβ-(α+β)γ+ $γ^{2}$ ][αβ-(α+β)δ+ $δ^{2}$ ]

(-b+aγ+ $γ^{2}$ )(-b+aδ+ $δ^{2}$ )

=(-b-b)(-b-b)=4 $b^{2}$

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Similar questions

If ∝ and β are roots of $x^{2}$ + ax - b = 0 and γ, δ arethe roots of $x^{2}$ + ax + b = 0, then (α-γ) (β-δ) (α-δ) (β-γ)

Q. If

α, β

are the roots of

x^{2} + p x + q = 0

and

γ, δ

are the roots of

x^{2} + p x - r = 0

, then

\frac{α - γ}{β - γ} \cdot \frac{α - δ}{β - δ}

is equal to

Q. If

α

and

β

are roots of

x^{2} + p x + 1 = 0

, and

γ

and

δ

are the roots of

x^{2} + q x + 1 = 0

show that

q^{2} - p^{2} - (α - γ) (β - γ) (α + δ) (β + δ) = 0

Q. If

α, β

are the roots of

x^{2} - 3 x + a = 0

and

γ, δ

that of

x^{2} - 12 x + b = 0

and

α, β, γ, δ

form an increasing G.P, then

Q. If

α, β

are the roots of

x^{2} - 8 x + A = 0

and

γ, δ

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x^{2} - 72 x + B = 0,

α < β < γ < δ

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If ∝ and β are roots of x2 + ax - b = 0 and γ, δ​ arethe roots of x2 + ax + b = 0, then (α-γ) (β-δ) (α-δ) (β-γ)

The correct option is A 4b2 α+β= -a , αβ= -b , γ+δ=-a γδ=b therefore (α-γ) (β-δ) (α-δ) (β-γ) =[ αβ-(α+β)γ+γ2][αβ-(α+β)δ+δ2] (-b+aγ+γ2)(-b+aδ+δ2) =(-b-b)(-b-b)=4b2

If ∝ and β are roots of $x^{2}$ + ax - b = 0 and γ, δ arethe roots of $x^{2}$ + ax + b = 0, then (α-γ) (β-δ) (α-δ) (β-γ)

The correct option is A
4 $b^{2}$

α+β= -a , αβ= -b , γ+δ=-a γδ=b

therefore (α-γ) (β-δ) (α-δ) (β-γ)

=[ αβ-(α+β)γ+ $γ^{2}$ ][αβ-(α+β)δ+ $δ^{2}$ ]

(-b+aγ+ $γ^{2}$ )(-b+aδ+ $δ^{2}$ )

=(-b-b)(-b-b)=4 $b^{2}$