Question

The roots of the equation are $l{x}^2+nx+n=0$ are in the ratio $p:q$. Then

• A. $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$
• B. $\sqrt{\frac{p}{q}}-\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$
• C. $\sqrt{\frac{p}{q}}-\sqrt{\frac{q}{p}}-\sqrt{\frac{n}{l}}=0$
• D. $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=1$

Answer 1

The correct option is A $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$

$l{x}^2+nx+n=0$

Roots are in ratio $p:q$

Let roots be $ap,aq$

$ap+pq=\frac{-n}{l},a^{2}pq=\frac{n}{l}$

$a=\sqrt{\frac{n}{pql}}$

$\therefore (p+q)\sqrt{\frac{n}{pql}}=\frac{-n}{l}$

$\sqrt{\frac{n}{l}}\left(\frac{p+q}{\sqrt{pq}}\right)=-{\left(\sqrt{\frac{n}{l}}\right)}^2$

$\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}=-\sqrt{\frac{n}{l}}$

$\therefore \sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$