CBSE Class 10 Answered

 if the roots are equal , then discriminant   b^2 - 4ac = 0 

           ( c- a) ^2  -  4 ( b-c ) ( a-b) = 0

              c^2 + a^2  - 2 ac  - 4 [ ba-b^2  -ca +bc]  = 0

             c^ 2  + a^2 + 2ac + 4b^2 -4ab - 4bc = 0

              [ c + a] ^2  = 4b [ a +c -b ]  

              if  a, b,c  are in A.P  then  c+a  = 2b

           substituting this  c+a = 2b in the above equation , we will get  [ c+a] ^2  = 4b[a+c -b] becomes 

          [  2b ]^2 = 4b[ 2b - b]  = 4b [ b] = 4b^2  THE CONDITION SATISFIES.

  therefore  a, b,c  are in A.P