Question

If the ratio of the roots of $l{x}^2+nx+n=0$ is $p:q$, then

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

Answer 1

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

Let the roots of $l{x}^2+nx+n=0$ be $\alpha \text{and}\beta$.
Then $\alpha +\beta =\frac{-n}{l},\alpha \beta =\frac{n}{l},\text{and}\frac{\alpha }{\beta }=\frac{p}{q}$
Now taking, $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }=\frac{\alpha +\beta +\alpha \beta }{\sqrt{\alpha \beta }}=\frac{-\frac{n}{l}+\frac{n}{l}}{\sqrt{\frac{n}{l}}}=0$