If the roots of the equation \(a x^2+a x+c=0\) are in the ratio \(p: q\), then…
If the roots of the equation \(a x^2+a x+c=0\) are in the ratio \(p: q\), then \(\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}=\)
\(\sqrt{\frac{a^2}{c}}\)
\(\sqrt{\frac{a}{2 c}}\)
\(\sqrt{\frac{a}{c}}\)
\(\sqrt{\frac{a^2}{2 c}}\)
Solution
Equation given be,
\(\begin{aligned}
a x^2+a x+c & =0 \\
\therefore \text { Sum of roots }=-\frac{a}{a} & =-1
\end{aligned}\)
\(\therefore\) Sum of roots \(=-\frac{a}{a}=-1\)
and product of roots \(=\frac{c}{a}\).
Now roots are given in ratio \(\frac{p}{q}\),
i.e., \(\frac{\alpha}{\beta}=\frac{p}{q}\)
So,
\(\begin{aligned}
\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}} & =\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}} \\
& =\frac{\alpha+\beta}{\sqrt{\alpha \beta}}=\frac{-1}{ \pm \sqrt{\frac{c}{a}}}=-\sqrt{\frac{a}{c}} \text { or } \sqrt{\frac{a}{c}}
\end{aligned}\)