If the ratio of roots of $ax^2+bx+c = 0\space$and $px^2+qx+r = 0\space$is same.
How to find ratio of their discriminants?
I don't understand this problem,what exactly is meant by ratio of the roots being same?
Let, $\alpha, \beta$ and $\gamma,\delta$ are the roots of the two equations respectively,does this problem says that $\frac{\alpha}{\beta} = \frac{\gamma}{\delta}=k$, where $k \in \mathbb{Q}$?
Even so I am not really much ideas how to continue without messing with tedious algebraic manipulation,again,considering this problem is of quantitative aptitude category,it may not be the right approach.Any ideas?
Best Answer
The ratio of the roots of the first quadratic polynomial are $(|b|-\sqrt{b^2-4ac})/(|b|+\sqrt{b^2-4ac})$. This is a one-to-one function of $ac/b^2$ hence the ratios coincide for the two polynomials if and only if $ac\cdot q^2=pr\cdot b^2$ $(*)$.
The discriminants of the quadratic forms are $D=b^2-4ac$ and $\Delta=q^2-4pr$ hence $(*)$ is equivalent to the condition that $b^2\cdot\Delta=q^2\cdot D$.