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

\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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None of these

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Solution

The correct option is A $\frac{a^{2}}{p^{2}}$
Let $A = α + h$ and $B = β = h,$ then $A - B = (α + h) - (β + h) = α - β$
$\Rightarrow {(A - B)}^{2} = {(α - β)}^{2} \Rightarrow {(A + B)}^{2} - 4 A B = {(α + β)}^{2} - 4 α β$
$\Rightarrow \frac{b^{2}}{a^{2}} - \frac{4 c}{a} = \frac{q^{2}}{p^{2}} - \frac{4 r}{p} \Rightarrow \frac{b^{2} - 4 a c}{a^{2}} = \frac{q^{2} - 4 r p}{p^{2}} \Rightarrow \frac{D_{1}}{a^{2}} = \frac{D_{2}}{p^{2}} \Rightarrow \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}}$

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