Question

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

• A. $n{a}^2=ac(n+1{)}^2$
• B. $n{c}^2=ab(n+1{)}^2$
• C. None of the above
• D. $n{a}^2=bc(n+1{)}^2$

Answer 1

The correct option is A

$n{a}^2=ac(n+1{)}^2$

Given: One root of the equation $a{x}^2+bx+c=0$ be $n$ times the other root.
Let the roots are $\alpha$ and $n\alpha$
Using, sum of roots
$\alpha +n\alpha =-\frac{b}{a}\Rightarrow \alpha =-\frac{b}{a(n+1)}....\left(i\right)$

and product,
$\alpha .n\alpha =\frac{c}{a}\Rightarrow {\alpha }^2=\frac{c}{na}....\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$, we get

$\Rightarrow {[-\frac{b}{a(n+1)}]}^2=\frac{c}{na}\Rightarrow \frac{b^2}{a^2(n+1{)}^2}=\frac{c}{na}$

$\Rightarrow n{b}^2=ac(n+1{)}^2$