Question

The ratio of the roots of the equation ${ax}^2+bx+c=0$ is same as the ratio of the roots of equation ${px}^2+qx+r=0$. If $D_1$ and $D_2$ are the discriminants of ${ax}^2+bx+c=0$ and ${px}^2+qx+r=0$ respectively, then $D_1:D_2=$

• A. $\frac{b^2}{q^2}$
• B. $\frac{b}{q}$
• C. $\frac{b}{q^2}$
• D. none of these

Answer 1

The correct option is A $\frac{b^2}{q^2}$

We have roots of a quadratic equation $a{x}^2+bx+c=0$ to be $\frac{-b\pm \sqrt{D_1}}{2a}$
And roots of $p{x}^2+qx+r=0$ to be $\frac{-q\pm \sqrt{D_2}}{2p}$.
So we have $\frac{-b+\sqrt{D_1}}{-b-\sqrt{D_1}}=\frac{-q+\sqrt{D_2}}{-q-\sqrt{D_2}}$, where $D_1$ is the discriminant of $a{x}^2+bx+c=0$ and $D_2$ is the discriminant of $p{x}^2+qx+r=0$ respectively.
Taking componendo dividendo we get,
$-\frac{\sqrt{D_1}}{b}=-\frac{\sqrt{D_2}}{q}$
$\Rightarrow \frac{D_1}{D_2}=\frac{b^2}{q^2}$