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Let abc be sides of triangle . No two are equal and € belongs to real numbers.If roots of x.x +2(a+b+c)x+3€(ab+bc+ca)=0 are real. Then range of€=?

Let abc be sides of triangle . No two are equal and € belongs to real numbers.If roots of x.x +2(a+b+c)x+3€(ab+bc+ca)=0 are real. Then range of€=?

Grade:11

2 Answers

Vikas TU
14149 Points
7 years ago
For roots to be real its Discriminant should be greater than equal to zero.
It should not be imaginary.
Hence,
4(a+b+c)^2 – 12€(ab+bc+ca) greater than equal to zero.
 
mycroft holmes
272 Points
7 years ago
In triangles we have the inequality
ab+bc+ca \le a^2+b^2+c^2 \le 2(ab+bc+ca) which can be written as
 
3(ab+bc+ca) \le (a+b+c)^2 \le 4(ab+bc+ca)
The left inequality arises from (a-b)^2+(b-c)^2+(c-a)^2 \ge 0 and the right inequality from the triangle inequalities squared i.e.
 
(a-b)^2<c^2 \\ \ (b-c)^2 <a^2 \\ \ (c-a)^2 <b^2
and summing up.
If (a+b+c)^2 - 3 \epsilon (ab+bc+ca) \ge 0 then the quadratic has real roots
 
So, we need \epsilon \le 1 if it has to be true for any value of a,b,c
 
Also, if \epsilon \ge \frac{4}{3} then it surely has imaginary roots. For 1 \le \epsilon \le \frac{4}{3} it is ambiguous whether the roots are real or imaginary.

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