Question

The condition for the equation $a{x}^2+bx+c=0$ to have one root $n$ times the other, is:

• A. $n{a}^2=bc{(n+1)}^2$
• B. $n{b}^2=ac{(n+1)}^2$
• C. $n{b}^2=ac{(n-1)}^2$
• D. None of these

Answer 1

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

Given that the roots of the equation $a{x}^2+bx+c=0$ be such that one root is $n$ times the other.
Let one root be $\alpha$, then the other root will be $n\alpha$ by given condition.
Sum of roots $=$ $S=\alpha +n\alpha =-\frac{b}{a}$
$\Rightarrow \alpha =-\frac{b}{a(1+n)}$.....(1)
Product of roots $=n{\alpha }^2=\frac{c}{a}$
$\Rightarrow {\alpha }^2=\frac{c}{an}$ ....(2)
From (1) and (2), we have
$\Rightarrow \frac{c}{an}=\frac{b^2}{a^2{(1+n)}^2}$
$\Rightarrow \therefore n{b}^2=ac{(n+1)}^2$