if a:b:c=1:m:n, where a,b,c are real numbers, and the expressions $x^{2} + 2 (a + b + c) x + 3 (b c + c a + a b)$ is a perfect square, then the value of m+n is

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Solution

The given expression is a perfect square
$\Rightarrow$ Discriminant=0
$\Rightarrow (2 (a + b + c))^{2} - 4 (3) (a b c + c a + a b) = 0$
$\Rightarrow (a + b + c))^{2} - 3 b c - 3 c a - 3 a b = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a - 3 b c - 3 c a - 3 a b$
$\Rightarrow a^{2} + b^{2} + c^{2} - a b - c a - b c = 0$
$\Rightarrow$ $\frac{1}{2}$ $[(a - b)^{2} + (b - c)^{2} + (c - a)^{2}] = 0$
This is possible only when a=b=c
$∴ a : b : c = 1 : 1 : 1$
$= 1 : m : n (g i v e n)$
or m=1, n=1 $\Rightarrow$ m+n=2

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