Here we can use the formula

$$\begin{gathered} \text{Average (A)=}\frac{\mathrm{Sum} \mathrm{of} \mathrm{observations} \left(\mathrm{S}\right)}{\mathrm{No}. \mathrm{of} \mathrm{observations} \left(\mathrm{N}\right)} \\ \text{⇒ S=A×N} \end{gathered}$$

According to the question,

Sum of weight of 21 boys (A=64, N=21)

⇒S=64 X 21 =1344

Later, weight of the teacher added, the average increased by 1 kg. (so A=65, N=22)

So, sum of weight of 22

⇒S=65 X 22 =1430

Now,

Weight of the teacher =(sum of wieght of 21 boys)-(Sum of weight of 22 )

=1430-1344

= 86 kg.

Hence, Option A is correct.