Question

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

• A. $-{(\iota +m)}^{2}pq=\iota m{r}^2$
• B. ${(\iota +m)}^{2}pr=\iota mq$
• C. ${(\iota +m)}^{2}pr=\iota m{q}^2$
• D. None of these

Answer 1

The correct option is A $-{(\iota +m)}^{2}pq=\iota m{r}^2$

For the equation $p{x}^2+qx-r=0,$

Let the roots be $ix$ and $mx$

Product of roots $=\frac{-r}{p}=(i+m)x$

$\Rightarrow x^2=[\frac{-r}{p(i+m)}{]}^2$ ...(1)

Sum of roots $=\frac{-q}{p}=im{x}^2$

$\Rightarrow x^2=\frac{-q}{imp}$ ...(2)

Equating (1) and (2), we get

$\frac{r^2}{p^2(i+m{)}^2}=-\frac{q}{imp}$

$\Rightarrow r^{2}im=-qp(i+m{)}^2$