If alpha, beta are the roots of ax^{2}+bx+c=0 | vedclass

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(A) Given, $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$.For the first equation, the discr

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

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(A) Given, $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$.For the first equation, the discriminant is $D_{1} = b^{2}-4ac$ and the difference of roots is $(\alpha-\beta)^{2} = \frac{D_{1}}{a^{2}}$.For the second equation, the discriminant is $D_{2} = q^{2}-4pr$ and the difference of roots is $((\alpha+h)-(\beta+h))^{2} = \frac{D_{2}}{p^{2}}$.Since $(\alpha+h)-(\beta+h) = \alpha-\beta$, we have $(\alpha-\beta)^{2} = ((\alpha+h)-(\beta+h))^{2}$.Therefore, $\frac{D_{1}}{a^{2}} = \frac{D_{2}}{p^{2}}$.This implies $\frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}}$.Thus, the ratio of the discriminants is $a^{2}: p^{2}$.

If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ $(a \neq 0)$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$ $(p \neq 0),$ then the ratio of the squares of their discriminants is

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