Mathematics - Quadratic Equation Question with Solution | TestHub

Given $\alpha ,\beta ,\gamma ,\delta$ are in A.P., let them be $\alpha ,\alpha +d,\alpha +2d,\alpha +3d$.

For $a{x}^2+bx+c=0$, roots are $\alpha ,\beta$.

Difference of roots: $|\alpha -\beta |=|\alpha -(\alpha +d)|=|-d|=|d|$.

Also, $|\alpha -\beta |=\frac{\sqrt{D_1}}{|a|}$.

So, $|d|=\frac{\sqrt{D_1}}{|a|}⟹d^2=\frac{D_1}{a^2}$.

For $p{x}^2+qx+r=0$, roots are $\gamma ,\delta$.

Difference of roots: $|\gamma -\delta |=|(\alpha +2d)-(\alpha +3d)|=|-d|=|d|$.

Also, $|\gamma -\delta |=\frac{\sqrt{D_2}}{|p|}$.

So, $|d|=\frac{\sqrt{D_2}}{|p|}⟹d^2=\frac{D_2}{p^2}$.

Equating the expressions for $d^2$:

$$\frac{D_1}{a^2}=\frac{D_2}{p^2}$$

Therefore, $D_1:D_2=a^2:p^2$.