Question

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

Answer 1

Answer: (B)

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

The equation is $l{x}^2+nx+n=0$
Let $\alpha$ and$\beta$ be the roots of the equation
Therefore
$\alpha +\beta =-n/l$
$\alpha \beta =n/l$
Given that $\alpha /\beta =p/q$ ...(i)
$\sqrt{n/l}-\sqrt{n/l}$
$=\sqrt{n/l}-\left[\right(n/l\left)/\sqrt{n/l}\right]$
$=\sqrt{n/l}+[\alpha +\beta /\sqrt{\alpha \beta }]$
$=\sqrt{n/l}+[\sqrt{p/q}+\sqrt{q/p}]$ ... from (i)
=0
Hence the answer is B