Question

If the roots of the equation px^2 + qr + r=0 are in the ratio l:m. Then, prove that (l+m)(l+m)pr=lmq.q

Answer 1

px2 +qr? +r =0

- so i think there ??? you have missed one x or ???

Answer 2

- and what is there (l+m)(l+m)pr=lmq ??? q

Answer 3

Sorry, the second term is qx.

Answer 4

ok and there ???

Answer 5

Let the roots of the equation be \[\alpha, \beta\]Then we know the roots are in the ratio,\[\frac{\alpha}{\beta}=\frac{l}{m} \rightarrow \alpha = \beta \frac{l}{m}\]We also know that the roots are related to the coefficients by,\[\alpha + \beta = -\frac{q}{p}\rightarrow \beta\frac{l}{m} +\beta=-\frac{q}{p}\rightarrow \beta=-\frac{q}{p}.\frac{m}{l+m}\]and\[\alpha \beta = \frac{r}{p} \rightarrow \left( \beta \frac{l}{m} \right) \beta= \frac{r}{p} \rightarrow \beta ^{2}=\frac{mr}{lp}\]Square the second-last result and equate with the last to find what you need,\[\frac{q^{2}}{p^{2}}\frac{m^{2}}{(l+m)^{2}}=\frac{mr}{lp}\ \rightarrow (l+m)^{2}pr=lmq^{2}\]