Question

If the roots of the equation $l{x}^2+nx+n=0$ be in the ratio p : q then $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=$

• A. $2\sqrt{\frac{n}{l}}$
• B. $\frac{n}{l}$
• C. None of these

Answer 1

Let roots of $l{x}^2+nx+n=0$ are $\alpha$ and $\beta$, given that $\frac{\alpha }{\beta }=\frac{p}{q}\Rightarrow \alpha =\beta \left(\frac{p}{q}\right)$
$\alpha +\beta =\frac{-n}{l}and\alpha \beta =\frac{n}{l}$
Now $\sqrt{\left(\frac{p}{q}\right)}+\sqrt{\left(\frac{q}{p}\right)}+\sqrt{\left(\frac{n}{l}\right)}=\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }$
$=\frac{(\alpha +\beta )-(\alpha +\beta )}{\sqrt{\alpha \beta }}=0$