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Question

if a:b:c=1:m:n, where a,b,c are real numbers, and the expressions x2+2(a+b+c)x+3(bc+ca+ab) is a perfect square, then the value of m+n is


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Solution

The given expression is a perfect square
Discriminant=0
(2(a+b+c))24(3)(abc+ca+ab)=0
(a+b+c))23bc3ca3ab=0
a2+b2+c2+2ab+2bc+2ca3bc3ca3ab
a2+b2+c2abcabc=0
12[(ab)2+(bc)2+(ca)2]=0
This is possible only when a=b=c
a:b:c=1:1:1
=1:m:n(given)
or m=1, n=1 m+n=2


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