If a = b = c, then the roots of the equation (x – a) (x – b) + (x

1.	If a = b = c, then the roots of the equation (x – a) (x – b) + (x – b)(x – c) + (x – c) (x – a) = 0 are A) imaginary B) real and equal C) real and unequal D) real
Answer» Correct option is (B) real and equal Given equation is (x – a) (x – b) + (x – b)(x – c) + (x – c) (x – a) = 0 \(\Rightarrow x^2-(a+b)x+ab+x^2-(b+c)x\) \(+bc+x^2-(a+c)x+ac=0\) \(\Rightarrow3x^2-((a+b)+(b+c)\) \(+(c+a))x+ab+bc+ac=0\) \(\Rightarrow3x^2-2(a+b+c)x+ab+bc+ac=0\) \(\Rightarrow3x^2-2\times3ax+a^2+a^2+a^2=0\) \((\because a=b=c)\) \(\Rightarrow3x^2-6ax+3a^2=0\) \(\Rightarrow x^2-2ax+a^2=0\) \(\Rightarrow(x-a)^2=0\) \(\Rightarrow\) x - a = 0 or x - a = 0 \(\Rightarrow\) x = a or x = a Hence, roots of given equation are real and equal. Correct option is B) real and equal

If a = b = c, then the roots of the equation (x – a) (x – b) + (x – b)(x – c) + (x – c) (x – a) = 0 are A) imaginary B) real and equal C) real and unequal D) real

Answer»

Correct option is (B) real and equal

Given equation is

(x – a) (x – b) + (x – b)(x – c) + (x – c) (x – a) = 0

\(\Rightarrow x^2-(a+b)x+ab+x^2-(b+c)x\) \(+bc+x^2-(a+c)x+ac=0\)

\(\Rightarrow3x^2-((a+b)+(b+c)\) \(+(c+a))x+ab+bc+ac=0\)

\(\Rightarrow3x^2-2(a+b+c)x+ab+bc+ac=0\)

\(\Rightarrow3x^2-2\times3ax+a^2+a^2+a^2=0\) \((\because a=b=c)\)

\(\Rightarrow3x^2-6ax+3a^2=0\)

\(\Rightarrow x^2-2ax+a^2=0\)

\(\Rightarrow(x-a)^2=0\)

\(\Rightarrow\) x - a = 0 or x - a = 0

\(\Rightarrow\) x = a or x = a

Hence, roots of given equation are real and equal.

Correct option is B) real and equal