Question

If one root is $n$ times the other for the equation $a{x}^2+bx+c=0,$ then

• A. $n{b}^2=ac(n+1{)}^2$
• B. $n^{2}b=ac(n+1{)}^2$
• C. $(nb{)}^2=a^{2}c(n+1)$
• D. $nb=ac(n+1)$

Answer 1

The correct option is A $n{b}^2=ac(n+1{)}^2$

Let the roots be $\alpha ,n\alpha .$
$\therefore \alpha +n\alpha =-\frac{b}{a}$
$\Rightarrow \alpha =-\frac{b}{a(1+n)}\cdots \left(1\right)$
and
$\therefore \alpha \cdot n\alpha =\frac{c}{a}$
$\Rightarrow {\alpha }^2=\frac{c}{an}\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$
$\Rightarrow \frac{b^2}{a^2(1+n{)}^2}=\frac{c}{an}$
$\Rightarrow n{b}^2=ac(n+1{)}^2$