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January 4, 2017

Condition of real roots | Tomato objective 291

Problem: If the roots of the equation [{(x-a)(x-b)}+{(x-b)(x-c)}+{(x-c)(x-a)}=0],(where a,b,c are real numbers) are equal , then

(A) [b^2-4ac=0]

(B) [a=b=c]

(C)  [a+b+c=0]

(D)    none of  foregoing statements is correct

 

Answer: ans (B) 

[{(x-a)(x-b)}+{(x-b)(x-c)}+{(x-c)(x-a)}=0]

[=>x^2-{(a+b)}x+ab+x^2-{(b+c)}x+bc+x^2-{(c+a)}x+ca=0]

[=>3x^2-2{(a+b+c)}x+(ab+bc+ca)=0]

discriminant, of the equation is

[4{(a+b+c)^2}-4.3{(ab+bc+ca)}=o]

[=>a^2+b^2+c^2+2(ab+bc+ca)-3(ab+bc+ca)=0]

[=>a^2+b^2+c^2-(ab+bc+ca)=0]

[=>a=b=c]

So, option (B) is correct ....

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