Question

If the roots of quadratic equation ax2+cx+c=0 are in ratio p:q then show that $\sqrt{ \frac{p}{q } } + \sqrt{ \frac{q}{p} } + \sqrt{ \frac{c}{a} } = 0$​

Answer 1

To prove: $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{c}{a}}=0$

Given quadratic equation,

           $ax^{2}+cx+c=0$

           Roots are in the ratio p:q.

           Let the roots be pk, qk.

           sum of the roots = -c/a

                 pk + qk = -c/a

                 p + q = (-c/a)(1/k)   --------------(1)

           product of roots = c/a

                  pk.qk = c/a

                  $pq=\frac{c}{a}.\frac{1}{k^{2} }$

             applying root on both sides

                  $\sqrt{pq}=\sqrt{ \frac{c}{a}}.\frac{1}{k}$        ----------------(2)

            Dividing (1) by (2)

                 $\frac{p+q}{\sqrt{pq} }=\frac{(-c/a)(1/k)}{\sqrt{\frac{c}{a} }(1/k) }$

                 $\frac{p}{\sqrt{pq} }+\frac{q}{\sqrt{pq} }=-\sqrt{\frac{c}{a}}.\sqrt{\frac{c}{a}}/\sqrt{\frac{c}{a}}$

                  $\frac{\sqrt{p}\sqrt{p} }{\sqrt{pq}}+\frac{\sqrt{q}\sqrt{q} }{\sqrt{pq}}=-\sqrt{\frac{c}{a} }$

                  $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}=-\sqrt{\frac{c}{a}}$

                  $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{c}{a}}=0$