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:

\frac{a^{2}}{p^{2}}

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\frac{a^{2}}{b^{2}}

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\frac{b^{2}}{q^{2}}

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\frac{c^{2}}{r^{2}}

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Solution

The correct option is A $\frac{a^{2}}{p^{2}}$
$α, β$ are roots of $a x^{2} + b x + c = 0$
$∴ α + β = \frac{- b}{a}, α β = \frac{c}{a}$
$D_{1} = b^{2} - 4 a c = a^{2} (\frac{b^{2}}{a^{2}} - \frac{4 c}{a})$
$= a^{2} ({(α + β)}^{2} - 4 α β)$
$= a^{2} {(α - β)}^{2}$
$γ, δ$ are roots of $p x^{2} + q x + r = 0$
$γ + δ = \frac{- q}{p}, γ δ = \frac{r}{p}$
$D_{2} = q^{2} - 4 p r = p^{2} (\frac{q^{2}}{p^{2}} - \frac{4 r}{p})$ $= p^{2} ({(γ + δ)}^{2} - 4 γ δ)$
$= p^{2} ({(γ - δ)}^{2})$
$α, β, γ, δ$ are in A.P.
$∴ α - β = γ - δ$
$∴ D_{1} : D_{2} = a^{2} : p^{2} \Rightarrow \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}}$

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