If - - are the roots of a x2-b x-c-0 and -k- -k are the roots of p x2-q x-r-0- then b2-4 a c-a2-4 p r is equal to

The correct option is B α + β = b/a, αβ = c/a ⇒ [(α + k)+(β + k)^2 4(α + k)(β + k)] = (α + β)2 4 αβ ⇒q^2/p^2 4 r/p = b^2/a^2 4c/a ⇒q^2 4pr/p^2=b^2 ...

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Standard XII

Mathematics

Range

If α, β are t...

Question

If $α, β$ are the roots of $a x^{2} + b x + c = 0 a n d α + k, β + k$ are the roots of $p x^{2} + q x + r = 0$ , then $\frac{b^{2} - 4 a c}{a^{2} - 4 p r}$ is equal to

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Solution

The correct option is C

$α + β = - \frac{b}{a}, α β = \frac{c}{a} \Rightarrow [{(α + k) + (β + k)}^{2} - 4 (α + k) (β + k)] = (α + β) 2 - 4 α β \Rightarrow \frac{q^{2}}{p^{2}} - 4 \frac{r}{p} = \frac{b^{2}}{a^{2}} - \frac{4 c}{a} \Rightarrow \frac{q^{2} - 4 p r}{p^{2}} = \frac{b^{2} - 4 a c}{a^{2}} \Rightarrow \frac{b^{2} - 4 a c}{q^{2} - 4 p r} = {(\frac{a}{p})}^{2}$

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If α,β are the roots of ax2+bx+c=0 and α+k, β+k are the roots of px2+qx+r=0, then b2−4aca2−4pr is equal to

The correct option is C α+β=−ba, αβ=ca⇒[(α+k)+(β+k)2−4(α+k)(β+k)]=(α+β)2−4αβ⇒q2p2−4rp=b2a2−4ca⇒q2−4prp2=b2−4aca2⇒b2−4acq2−4pr=(ap)2

If $α, β$ are the roots of $a x^{2} + b x + c = 0 a n d α + k, β + k$ are the roots of $p x^{2} + q x + r = 0$ , then $\frac{b^{2} - 4 a c}{a^{2} - 4 p r}$ is equal to