If - are the roots of ax2 - bx - c - 0 and - - k- - - k are the roots of px2 - qx - r - 0-Then b2 - 4acq2 - 4pr is equal to

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Transformation of Roots: Linear Combination of Roots

If α,β are th...

Question

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

$1$

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

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$0$

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

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Solution

The correct option is B
${(\frac{a}{p})}^{2}$

Given: $α, β$ are the roots of $a x^{2} + b x + c = 0$ and $α + k, β + k$ are the roots of $p x^{2} + q x + r = 0$ .

Absolute difference of both the roots will be same,
$| α - β | = | α + k - β - k | \Rightarrow [{(α + k + β + k)}^{2} - 4 (α + k) (β + k)] = (α + β)^{2} - 4 α β \Rightarrow \frac{q^{2}}{p^{2}} - 4 \frac{r}{p} = \frac{b^{2}}{a^{2}} - \frac{4 c}{a} \Rightarrow \frac{q^{2} - 4 p r}{p^{2}} = \frac{b^{2} - 4 a c}{a^{2}} \Rightarrow \frac{b^{2} - 4 a c}{q^{2} - 4 p r} = {(\frac{a}{p})}^{2}$

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Similar questions

If $\alpha,\beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + k,~\beta + k$ are the roots of $px^2 + qx + r = 0$.
Then $\dfrac{b^2 - 4ac}{q^2 - 4pr}$ is equal to

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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 b2−4acq2−4pr is equal to

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