If $α, β$ are the roots of $a x^{2} + b x + c = 0$ ; $α + h, β + h$ are the roots of $p x^{2} + q x + r = 0$ ; and $D_{1}, D_{2}$ are the respective discriminants of these equations, then $D_{1} : D_{2}$ is

\frac{a^{2}}{p^{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 $\frac{a^{2}}{p^{2}}$
$α + β = \frac{- b}{a}$
$α β = \frac{c}{a}$
$α + h + β + h = \frac{- q}{p}$
$(α + h) (β + h) = \frac{r}{p}$
$D_{1} = b^{2} - 4 a c$
$D_{2} = q^{2} - 4 p r$
$(α - β)^{2} = \frac{b^{2}}{a^{2}} - \frac{4 c}{a} = \frac{q^{2}}{p^{2}} - \frac{4 r}{p}$
$D_{1} = (α - β)^{2} a^{2}$
$D_{2} = (α - β)^{2} p^{2}$
$∴ \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}}$

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$α, β$ 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 $D_{1} : D_{2} = (w h e r e α, β, γ δ, ϵ R & a, b, c, p, q, r ϵ R)$