Question

If the ratio of the roots of the equation $a{x}^2+bx+c=0$ be $p:q$, then

• A. $pq{b}^2+(p+q{)}^{2}ac=0$
• B. $pq{b}^2-(p+q{)}^{2}ac=0$
• C. $pq{a}^2-(p+q{)}^{2}bc=0$
• D. $pq{b}^2-(p-q{)}^{2}ac=0$

Answer 1

The correct option is B

$pq{b}^2-(p+q{)}^{2}ac=0$

Let pα,qα be the roots of the given equation $a{x}^2+bx+c=0$

Then pα + qα = $-\frac{b}{a}andp\alpha .q\alpha =\frac{c}{a}$

From first relation, α = $-\frac{b}{a(p+q)}$

Substituting this value of α in second relation, we get

$\frac{b^2}{a^2(p+q{)}^2}\times pq=\frac{c}{a}$ ⇒ $b^{2}pq-ac(p+q{)}^2=0$

Note : Students should remember this question as a fact.