If - are the roots of quadratic equation x2-p x-q-0 and - are the roots of x2-px-r-0 then - is equal to-

The correct option is A (q+r) Given that, α, β #160; are the roots of the equation x^2 + px + q = 0 α + β = ba = p and αβ = ca ...

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Standard XII

Mathematics

Domain and Range of Trigonometric Ratios

If α,β are ...

Question

If $α, β$ are the roots of quadratic equation $x^{2} + p x + q = 0$ and $γ δ$ are the roots of $x^{2} + p x - r = 0$ then $(α - γ) . (α - δ)$ is equal to:

q + r

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q - r

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- (q + r)

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- (p + q + r)

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Solution

The correct option is A $- (q + r)$
Given that,
$α, β$ are the roots of the equation $x^{2} + p x + q = 0$

$⟹ α + β = \frac{- b}{a} = - p$ and $α β = \frac{c}{a} = q$

$γ, δ$ are the roots of the equation $x^{2} + p x + q = 0$

$⟹ γ + δ = \frac{- b}{a} = - p$ and $γ δ = \frac{c}{a} = - r$

$∴ (α - γ) (α - δ) = α^{2} - α (γ + δ) + γ δ$

$= α^{2} + p α - r$

$= - q - r$

$= - (q + r)$ (Ans)

$∵ α$ is a root of $x^{2} + p x + q = 0$

So, $α^{2} + p α + q = 0$

$α^{2} + p α = - q$

Option C is correct.

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If α,β are the roots of quadratic equation x2+p x+q=0 and γδ are the roots of x2+px−r=0 then (α−γ).(α−δ) is equal to:

If $α, β$ are the roots of quadratic equation $x^{2} + p x + q = 0$ and $γ δ$ are the roots of $x^{2} + p x - r = 0$ then $(α - γ) . (α - δ)$ is equal to: