Find the condition that roots of the equation ax² + bx + c = 0 are in the ratio m : n.

Let mα and nα be the roots of the equation ax² + bx + c = 0

Now, mα + nα = -b/a ……………….…(i)

and mnα² = c/a …………………….…(ii)

From (i): α(m + n) = -b/a

⇒ α²(m + n)² = b²/a²

Using (ii): (c/a)(m + n)² = mn(b²/a²)

⇒ ac(m + n)² = mnb²

Which is the required condition.