<s>[INST] Please, I need a more detailed explanation of the particular solution of the problem with vectors Here is the problem and its solution (link to the source if you are interested):
Two different points $A$ and $B$ are given. Find a set of such points $M$, that $\overrightarrow{MA}\cdot\overrightarrow{MB}=k^2$, where $k$ is a given non-zero number.
The solution:
Let coordinates of points be: $A(x_a, y_a), B(x_b, y_b), M(x, y)$.
Then: 
$$
\begin{align}
& \overrightarrow{MA}=(x_a - x, y_a - y), \overrightarrow{MB}=(x_b - x, y_b - y), \\ 
& \overrightarrow{MA}\cdot\overrightarrow{MB}= (x_a - x)(x_b - x)+(y_a - y)(y_b - y)=k^2
\end{align}
$$
Transform the last expression, opening brackets:
$$
x_a x_b - (x_a+x_b)x + x^2 + y_a y_b - (y_a+y_b)y + y^2 = k^2
$$
At this point I ask someone to provide a more detailed explanation, please, cause I failed to understand the next steps. What a transformation has been made here?
$$
\left( x - \frac{x_a + x_b}{2} \right)^2 - \frac{(x_a - x_b)^2}{4} + 
\left( y - \frac{y_a + y_b}{2} \right)^2 - \frac{(y_a - y_b)^2}{4} = k^2 \\
\left( x - \frac{x_a + x_b}{2} \right)^2 + \left( y - \frac{y_a + y_b}{2} \right)^2 
 = k^2 + \frac{(x_a - x_b)^2}{4} + \frac{(y_a - y_b)^2}{4}
$$
Thus, this is a circumference with a center at the middle of segment $AB$ and its radius. (This I also failed to understand)
$$
r = \sqrt{k^2 + \frac{|AB|^2}{4}}
$$
 [/INST] Are you familiar with how to "complete the square?"  To get from
$$
x_a x_b - (x_a+x_b)x + x^2 + y_a y_b - (y_a+y_b)y + y^2 = k^2
$$
to 
$$
\left( x - \frac{x_a + x_b}{2} \right)^2 - \frac{(x_a - x_b)^2}{4} + 
\left( y - \frac{y_a + y_b}{2} \right)^2 - \frac{(y_a - y_b)^2}{4} = k^2 \\
$$
consider the following: 
$$
\begin{align}
x_ax_b-(x_a+x_b)x+x^2&=x_ax_b-(x_a+x_b)x+x^2+\frac{(x_a - x_b)^2}{4}-\frac{(x_a - x_b)^2}{4}\\
&=\frac{4x_ax_b}{4}-(x_a+x_b)x+x^2+\frac{x_a^2-2x_ax_b+x_b^2}{4}-\frac{(x_a - x_b)^2}{4}\\
&=-(x_a+x_b)x+x^2+\frac{x_a^2+2x_ax_b+x_b^2}{4}-\frac{(x_a - x_b)^2}{4}\\
&=-(x_a+x_b)x+x^2+\frac{(x_a + x_b)^2}{4}-\frac{(x_a - x_b)^2}{4}\\
&=\left( x - \frac{x_a + x_b}{2} \right)^2-\frac{(x_a - x_b)^2}{4}.\\
\end{align}
$$
Similarly, 
$$
\begin{align}
y_ay_b-(y_a+y_b)y+y^2&=\left( y - \frac{y_a + y_b}{2} \right)^2-\frac{(y_a - y_b)^2}{4}.\\
\end{align}
$$
As for the second part of your question, the standard equation for a circle with center at $P(h,k)$ and radius $r$ is $(x-h)^2+(y-k)^2=r^2$.  Therefore, 
$$
\left( x - \frac{x_a + x_b}{2} \right)^2 + \left( y - \frac{y_a + y_b}{2} \right)^2 
 = k^2 + \frac{(x_a - x_b)^2}{4} + \frac{(y_a - y_b)^2}{4}
$$
is the equation for the circle with center at $$O\left(\frac{x_a+x_b}{2},\frac{y_a+y_b}{2}\right)$$ and radius 
$$\begin{align}
r&=\sqrt{k^2 + \frac{(x_a - x_b)^2}{4} + \frac{(y_a - y_b)^2}{4}}\\
&=\sqrt{k^2 + \frac{|AB|^2}{4}}.
\end{align}
$$
Note that the coordinates of $O$ are the coordinates for the midpoint of segment $AB$.
Hope this helps.
</s>