If α,β are the roots of ax2+ bx+ c =0 (a≠0) and α+h,β+h are the roots of px2+ qx+ r=0,(p≠0)then the ratio of the squares of their discriminants is

Given, cr and p are the roots of ai + br* c = 0 and ct, + h and 9 + hare the roots of p* + qr* r =0. ∴α+β=−ba and αβ=ca and α+h+β+h=−qp,(α+h)(β+h)=rpand α+h+β+h=−qp,(α+h)(β+h)=rpNow ,(α+h)−(β+h)=α−β⇒[(α+h)−(β+h)]2=(α−β)2⇒[(α+h)+(β+h)]2−4(α+h)(β+h) =(α+β)2−4αβ⇒ q2p2−4rp=b2a2−4ca⇒ q2−4prp2=b2−4aca2∴ b2−4acq2−4pr=a2p2Hence, the ratio of the squares of their discriminantsis a2 : p2.

If α,β are the roots of ax2+ bx+ c =0 (a≠0) and α+h,β+h are the roots of px2+ qx+ r=0,(p≠0)then the ratio of the squares of their discriminants is

Given, cr and p are the roots of ai + br* c = 0 and ct, + h and 9 + hare the roots of p* + qr* r =0. ∴α+β=−ba and αβ=ca and α+h+β+h=−qp,(α+h)(β+h)=rpand α+h+β+h=−qp,(α+h)(β+h)=rpNow ,(α+h)−(β+h)=α−β⇒[(α+h)−(β+h)]2=(α−β)2⇒[(α+h)+(β+h)]2−4(α+h)(β+h) =(α+β)2−4αβ⇒ q2p2−4rp=b2a2−4ca⇒ q2−4prp2=b2−4aca2∴ b2−4acq2−4pr=a2p2Hence, the ratio of the squares of their discriminantsis a2 : p2.

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If $α$ , $β$ are the roots of ax²+ bx+ c =0 $(a \neq 0)$ and $α + h, β + h$ are the roots of px²+ qx+ r=0, $(p \neq0)$ then the ratio of the squares of their discriminants is

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

$a : p^{2}$

$a^{2} : p$

a:2p

answer is A.

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Detailed Solution

Given, cr and p are the roots of ai + br* c = 0 and ct, + h and 9 + hare the roots of p* + qr* r =0.
$\begin{aligned} ∴ α + β = - \frac{b}{a} and αβ = \frac{c}{a} \\ and α + h + β + h = - \frac{q}{p}, (α + h) (β + h) = \frac{r}{p} \end{aligned}$
and $α + h + β + h = - \frac{q}{p}, (α + h) (β + h) = \frac{r}{p}$
Now , $(α + h) - (β + h) = α - β$
$\Rightarrow [(α + h) - (β + h)]^{2} = (α - β)^{2}$
$\Rightarrow [(α + h) + (β + h)]^{2} - 4 (α + h) (β + h)$
$= (α + β)^{2} - 4 αβ$
$\Rightarrow \frac{q^{2}}{p^{2}} - \frac{4 r}{p} = \frac{b^{2}}{a^{2}} - \frac{4 c}{a}$
$\Rightarrow \frac{q^{2} - 4 pr}{p^{2}} = \frac{b^{2} - 4 ac}{a^{2}}$
$∴ \frac{b^{2} - 4 ac}{q^{2} - 4 pr} = \frac{a^{2}}{p^{2}}$
Hence, the ratio of the squares of their discriminants
is a² : p².