Find the condition that the roots of the quadratic equation $ax^2+cx+c=0$ may be in the ratio of $m:n$
My Attempt
If $\alpha $ and $\beta $ are the roots of the equation $ax^2+cx+c=0$ then
$$\alpha + \beta = -\dfrac {c}{a}$$
$$\alpha.\beta =\dfrac {c}{a}$$
Also,
$$\dfrac {\alpha}{\beta }=\dfrac {m}{n}$$
using componendo and dividendo,
$$\dfrac {\alpha +\beta}{\alpha -\beta}=\dfrac {m+n}{m-n}$$
You can solve one of your equations for $\beta$: $$\beta=\frac{n}{m}\alpha$$
Now sub this into the sum and product equations: $$\alpha\left(1+\frac{n}{m}\right)=-\frac{c}{a}$$ and $$\frac{n}{m}\alpha^2=\frac{c}{a}$$ Let $k=\frac{c}{a}$, and you've got two equations with the variables $n,m,\alpha,k$. If you eliminate $\alpha$, you'll have one equation with just $m,n,k$. Solve it for $k$, and you'll know the required ratio of $c$ to $a$ in terms of $m$ and $n$.