Let $α, β$ are the roots $a x^{2} + b x + c = 0$ 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 $α, β, γ$ and $δ$ are in A.P then $D_{1} : D_{2} =$

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

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

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

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

answer is A.

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Detailed Solution

$α, β, γ, δ$ are in A.P

$β = α + d, γ = α + 2 d, δ = α + 3 d$

$d^{2} = {(β - α)}^{2} = {(α + β)}^{2} - 4 α β$

$= \frac{b^{2}}{a^{2}} - \frac{4. c}{a} = \frac{b^{2} - 4 a c}{a^{2}} = \frac{D_{1}}{a^{2}}$

$d^{2} = {(δ - γ)}^{2} = {(γ + δ)}^{2} - 4 γ δ$

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

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

$D_{1} : D_{2} = a^{2} d^{2} : p^{2} d^{2} = a^{2} : p^{2}$

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