Question

The condition that the roots of the equation $a{x}^2+bx+c=0$ be such that one root is $n$ times the other is

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

Answer 1

Answer: (B)

$n{b}^2=ca{(n+1)}^2$

Let $p$ and $np$ be the roots of the given equation $a{x}^2+bx+c=0$.

So, $p+np=-\frac{b}{a}$, and $n{p}^2=\frac{c}{a}$.

Consider,

$p+np=-\frac{b}{a}\Rightarrow p(n+1)=-\frac{b}{a}\Rightarrow p=-\frac{b}{a(n+1)}$

Therefore,

$n{p}^2=\frac{c}{a}\Rightarrow n{(-\frac{b}{a(n+1)})}^2=\frac{c}{a}\Rightarrow \frac{n{b}^2}{a^2(n+1{)}^2}=\frac{c}{a}\Rightarrow n{b}^{2}a=c{a}^2(n+1{)}^2\Rightarrow n{b}^2=\frac{c{a}^2(n+1{)}^2}{a}\Rightarrow n{b}^2=ca(n+1{)}^2$.