If - - are the roots of x2-px-q-0 and - - are the roots of x2-px-r-0- then -

The correct option is A 1 Since, α, β are the roots of x^2+ px+q =0 #160; α+β= p and αβ=q Also, γ, δ #160;are roots #160 ...

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Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

If α, β are...

Question

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 $\frac{(α - γ) (α - δ)}{(β - γ) (β - δ)} =$

1

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q + r

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Solution

The correct option is A $1$
Since, $α, β$ are the roots of $x^{2} + p x + q = 0$
$α + β = - p$ and $α β = q$
Also, $γ, δ$ are roots of $x^{2} + p x + r = 0$ ,
$γ + δ = - p$ and $γ δ = r$
Consider $\frac{(α - γ) (α - δ)}{(β - γ) (β - δ)}$
$= \frac{α^{2} - (γ + δ) α + γ δ}{β^{2} - (γ + δ) β + γ δ}$
$= \frac{α^{2} + p α + r}{β^{2} + p β + r}$
$= \frac{- q + r}{- q + r} (∵ α, β$ are roots of $x^{2} + p x + q = 0)$
$= 1$

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If α,β are the roots of x2+px+q=0 and γ,δ are the roots of x2+px+r=0, then (α−γ)(α−δ)(β−γ)(β−δ)=

The correct option is A 1Since, α,β are the roots of x2+px+q=0 α+β=−p and αβ=qAlso, γ,δ are roots of x2+px+r=0,γ+δ=−p and γδ=rConsider (α−γ)(α−δ)(β−γ)(β−δ)=α2−(γ+δ)α+γδβ2−(γ+δ)β+γδ=α2+pα+rβ2+pβ+r=−q+r−q+r(∵α,β are roots of x2+px+q=0)=1

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 $\frac{(α - γ) (α - δ)}{(β - γ) (β - δ)} =$