Question

If the ratio of the roots of $x^2+bx+c=0$ is equal to the ratio of the roots of $x^2+px+q=0$ then $p^{2}c-b^{2}q=$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

$x^2+bx+c=0$ $x^2+px+q=0$

roots $\alpha ,\beta$ roots $r,\delta$

$\frac{\alpha }{\beta }=\frac{r}{\delta }$ $\alpha +\beta =-b$ $r\delta =q$

$\alpha \beta =c$ $r+\delta =-p$

applying components & dividend

$\frac{\alpha +\beta }{\alpha -\beta }=\frac{r+\delta }{r-\delta }$

$\alpha -\beta =\sqrt{(\alpha +\beta {)}^2-4\alpha \beta }$

$\frac{\alpha +\beta }{\sqrt{(\alpha +\beta {)}^2-4\alpha \beta }}=\frac{r}{\sqrt{p^2-4q}}$

$b^2(o^2-4q)=p^2(b^2-4c)$

$b^2/p^2-4b^{2}q=p^2/b^2-4p^{2}c$

$4(p^{2}c-b^{2}q)=0$

$p^{2}c-b^{2}q=0$