Question

If the roots of the equation $p{x}^2+qx+r=0$ are in the ratio $\phi :m,$ then

• A. $(\phi +m{)}^{2}qp=\phi m{r}^2$
• B. $(\phi +m{)}^{2}pr=\phi mq$
• C. $(\phi +m{)}^{2}pr=\phi m{q}^2$
• D. None of the above

Answer 1

The correct option is C $(\phi +m{)}^{2}pr=\phi m{q}^2$

Let the roots be $\phi r$ and $mr$

Sum of roots $=\phi r+mr=-\frac{q}{p}$

$\therefore (\phi +m)r=-\frac{q}{p}$

$\therefore (\phi +m{)}^2{r}^2=\frac{q^2}{p^2}$ ...(1)

Product of roots $=\left(\phi r\right)\left(mr\right)=\frac{r}{p}$

$\therefore \phi m{r}^2=\frac{r}{p}$ ...(2)

Dividing equation (1) by (2), we get

$\frac{(\phi +m{)}^2{r}^2}{\phi m{r}^2}=\frac{\frac{q^2}{p^2}}{\frac{r}{p}}$

$\therefore \frac{(\phi +m{)}^2}{\phi m}=\frac{q^2}{pr}$

$\therefore (\phi +m{)}^{2}pr=\phi m{q}^2$