State true or false:Both the roots of the equation

x−a

x−b
+

x−b

x−c
+

x−c

x−a
=0
are real but they are equal only when a=b=c.

Question

State true or false:Both the roots of the equation

x−a

x−b
   +
    
     x−b

x−c
   +
    
     x−c

x−a
   =0
  are real but they are equal only when a=b=c.

Accepted Answer

Given equation is

(x−a)(x−b)+(x−b)(x−c)+(x−c)(x−a)=0
  Now,Compare given equation with general form,

a
    x
    2
   
   +bx+c=0

a

=3

b

=−2(a+b+c)

c

=ab+bc+ca

Assume D be the discriminant of quadratic equation.So,

D=
      (−2(a+b+c))
      2
     
     −4×3×(ab+bc+ca)

⇒D=4
      (a+b+c)
      2
     
     −12(ab+bc+ca)

⇒D=4

a
        2
       
       +
        b
        2
       
       +
        c
        2
       
       +2ab+2bc+2ca
     −12ab−12bc−12ca

⇒D=
      
       (4
        a
        2
       
       +4
        b
        2
       
       +4
        c
        2
       
       +8ab+8bc+8ca
     −12ab−12bc−12ca

The square of any number is always a positive number.From the discriminant it is clear that 2 is a positive number and

(a−b)
    2
   
   ,
    
     (b−c)
    2

and

(a−c)
    2

are greater than zero, so the product of a positive number and positive number is always a positive number it means the number is strictly greater than zero when

a≠b≠c
   .As the discriminant is strictly greater than zero, so the roots of given quadratic equation are real. When

a=b=c
   the discriminant becomes 0, so the roots are equal when

a=b=c
  Therefore, it is true that the roots of the equation

x−a

x−b
   +
    
     x−b

x−c
   +
    
     x−c

x−a
   =0
   are real but they are equal only when

a=b=c
  .Hence, the correct option is 1.

State true or false:

Both the roots of the equation $x − a x − b + x − b x − c + x − c x − a = 0$ are real but they are equal only when $a = b = c$ .

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State true or false:Both the roots of the equation x−a x−b + x−b x−c + x−c x−a =0 are real but they are equal only when a=b=c.