If the roots of $$ax^2 + bx + c = 0$$    are the reciprocals of those of $$\ell {x^2} + mx + n = 0$$    then $$a : b : c =$$

Solution :
If $$\alpha , \beta$$  be the roots then
$$\alpha + \beta = - \frac{b}{a},\alpha \beta = \frac{c}{a}$$
Now the roots of $$\ell x^2 + mx + n = 0$$    are $$\frac{1}{\alpha },\frac{1}{\beta }$$
$$\eqalign{ & \therefore \frac{1}{\alpha } + \frac{1}{\beta } = - \frac{m}{\ell }{\text{and}}\frac{1}{\alpha } \cdot \frac{1}{\beta } = \frac{n}{\ell } \cr & {\text{or, }}\frac{{\alpha + \beta }}{{\alpha \beta }} = - \frac{m}{\ell }{\text{and}}\frac{a}{c} = \frac{n}{\ell } \cr & {\text{or, }} - \frac{b}{c} = - \frac{m}{\ell }{\text{and}}\frac{a}{c} = \frac{n}{\ell } \cr & {\text{or, }}\frac{a}{n} = \frac{b}{m} = \frac{c}{\ell }. \cr & \therefore a:b:c = n:m:\ell \cr} $$