Simplification of given data
Given : a,b,c are in A.P.
⇒b−a=c−b
⇒b−a−c+b=0
⇒2b−a−c=0 ...(i)
Solve given determinant
△=∣∣
∣∣2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c∣∣
∣∣
Multiply & divide by 2
⇒△=22∣∣
∣∣2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c∣∣
∣∣
⇒△=12∣∣
∣
∣∣2y+45y+78y+a2(3y+5)2(6y+8)2(9y+b)4y+67y+910y+c∣∣
∣
∣∣
⇒△=12∣∣
∣∣2y+45y+78y+a6y+1012y+1618y+2b4y+67y+910y+c∣∣
∣∣
Applying R2→R2−R1−R3
⇒△=12∣∣
∣∣2y+45y+78y+a002b−a−c4y+67y+910y+c∣∣
∣∣
⇒△=12∣∣
∣∣2y+45y+78y+a0004y+67y+910y+c∣∣
∣∣ ....[from (i)]
⇒△=12×0
⇒△=0