Question

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

• A. $n{a}^2$ = bc$(n+1{)}^2$
• B. $n{b}^2$ = ac$(n+1{)}^2$
• C. $n{c}^2$ = ab$(n+1{)}^2$
• D. $n{b}^3$ = ac$(n+1{)}^2$

Answer 1

The correct option is B

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

Let the roots be $\alpha$ and n $\alpha$

Sum of roots, α + nα = -$\frac{b}{a}$ ⇒ α = -$\frac{b}{a(n+1)}$ ......(i)

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

From (i) and (ii), we get

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

⇒ $n{b}^2$ = ac$(n+1{)}^2$.

Note : Students should remember this question as a fact.