Given a point:
$$ \overline{x} = (x_1,x_2)$$
And a curve: $$ a_1(t) = (a_{11}(t), a_{12}(t)) $$
I am asked to find the function
$$ a_2(t) = (a_{21}(t), a_{22}(t))$$
Such that:
$$ (a_2(t)-\overline{x})\cdot(a_2(t)-\overline{x})=c_1\tag1$$ $$ (a_2(t)-a_1(t))\cdot(a_2(t)-a_1(t))=c_2\tag2$$
If we define $$F:R^3 \mapsto R, F(\overline{x},t) = (\overline{x}-a_1(t))\cdot (\overline{x}-a_1(t))\tag3$$ $$ h(t) = (a_{21}(t), a_{22}(t), t) \tag4$$
Since $ F \circ h(t) = c_2 $ using the chain rule:
$$ (a_1(t) \cdot a'_1(t)) + (a_2(t) \cdot a_2'(t))- (a_1'(t)\cdot a_2(t) )- (a'_2(t)\cdot a_1(t)) = 0\tag5$$
From here I suppose that $$ a_1(t) = (c_1) ^2(sin(k(t)) + x_1, cos(k(t)) + x_2)\tag6$$
And try t solve for $k(t)$. I dont have idea how to solve the problem, I need a hint or something because I dont think I am going in the right diretion.