Let x2 occur in the (r + 1)th term of the expansion (3+ax)9 . Then
Tr+1 =

9

‌

Cr(3)9−r(ax)r =

9

‌

Cr(3)9−rarxr
On comparing power of x in x2 and Tr+1, we get r = 2
⇒ T2+1 =

9

‌

C2(3)9−2a2x2
∴ Coefficient of x2 in T3 is

9

‌

C2(3)7a2 = 36 (3)7a2
Now let x3 occurs in the (r + 1)th term of the expansion (3+ax)9
∴ Tr+1 =

9

‌

Cr(3)9−r(ax)r =

9

‌

Cr(3)9−rarxr
On comparing power of x in x3 and Tr+1, we get r = 3
⇒ T3+1 =

9

‌

C3(3)9−3(ax)3 =

9

‌

C3(3)6a3x3
Coefficient of x3 in T4 is

9

‌

C336a3 = 84(3)6a3
We are given coefficient of x2 = coefficient of x3
∴ 36 (3))7‌a2 = 84 (3)6a3
⇒

a3

a2

=

36(3)7

84(3)6

=

9

7

⇒ a =

9

7