find the condition that the ratio between the roots of the eqn. ax^{2}+bx+c=0 may be m:n.

Let the roots be my & ny

Therefore sum of roots => my+ny = -b/a

(m+n)y = -b/a .......1

Product of roots=> my x ny = c/a

mny^{2}=c/a .........2

Squaring (1), we get

(m+n)^{2} y^{2 }= (-b/a )^{2 }...........3

Dividing 3 by 2…………(y^{2} will be cancelled)

(m+n)^{2}/mn = b^{2}/a^{2} x a/c

ac(m+n)^{2} = b^{2}mn.

Hence, the required condition is ac(m+n)^{2} = b^{2}mn.

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