If $α, β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0,$ $γ, δ$ are the roots of $p x^{2} + q x + r = 0 & D_{1}, D_{2}$ be the respective discriminants of these equations. If $α, β, γ, & δ$ are in A.P. then $D_{1} : D_{2} =$
Where $α, β, γ, δ, \in R & a, b, c, p, q, r \in R)$

$a^{2} : b^{2}$

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$c^{2} : r^{2}$

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$a^{2} : r^{2}$

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$a^{2} : p^{2}$

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Solution

The correct option is D
$a^{2} : p^{2}$

Given $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$
$\Rightarrow α + β = - \frac{b}{a}, α β = \frac{c}{a}$

Now, $D_{1} = b^{2} - 4 a c = a^{2} (\frac{b^{2}}{a^{2}} - \frac{4 c}{a})$

$= a^{2} ((α + β)^{2} - 4 α β)$

$= a^{2} (α - β)^{2}$

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

$\Rightarrow γ + δ = - \frac{q}{p}, γ δ = \frac{r}{p}$

$D_{2} = q^{2} - 4 p r = p^{2} (\frac{q^{2}}{p^{2}} - \frac{4 r}{p})$

$= p^{2} (γ + δ)^{2} - 4 γ δ$

$= p^{2} (γ - δ)^{2}$

$∵ α, β, γ, δ$ are in A.P

$∴ α - β = γ - δ$

$∴ \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}}$

Hence, $D_{1} : D_{2} = a^{2} : p^{2}$

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Similar questions

Q. Let

α, β

be the roots of

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

and

γ, δ

be the roots of

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

; and

D_{1}, D_{2}

the respective Discriminants of these equations. If

α, β, γ, δ

are in A.P, then

D_{1} : D_{2}

is:

$α, β$ are the roots of $a x^{2} + b x + c = 0 a n d γ, δ a r e t h e r o o t s o f p x^{2} + q x + r = 0 a n d D_{1}, D_{2}$ be the respective discriminants of these equations. If $α, β, γ, a n d δ$ are in A.P. then $D_{1} : D_{2} = (w h e r e α, β, γ δ, ϵ R & a, b, c, p, q, r ϵ R)$