In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.
To solve this problem, we need to maximize the ratio \(\frac{|BC|}{|AB|}\) for a line \(\ell\) parallel to \(b\). Let's break down the solution step by step.

1. **Define the Geometry:**
   - Let \(D\) be the center of the circle \(\omega\).
   - Let \(x\) be a line parallel to \(b\) passing through \(D\).
   - Let \(O = x \cap a\).
   - Let \(y\) be a line perpendicular to \(x\) passing through \(O\).
   - Let \(E\) be the point where \(x\) intersects the circle \(\omega\).
   - The circle \(\omega\) can be described by the equation \((x - a)^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

2. **Equation of Line \(a\):**
   - Assume the line \(a\) has the equation \(y = kx\).

3. **Intersection Points:**
   - Let \(\ell\) be a line parallel to \(b\) intersecting \(a\) at point \(A\) and the circle \(\omega\) at points \(B\) and \(C\).
   - Since \(\ell \parallel b\), it can be written as \(y = mx + c\) where \(m\) is the slope of \(b\).

4. **Distance Calculation:**
   - The distance \(BC\) is the chord length of the circle, which can be calculated using the formula for the chord length:
     \[
     BC = 2\sqrt{r^2 - y^2}
     \]
   - The distance \(AB\) can be calculated using the coordinates of \(A\) and \(B\).

5. **Ratio Maximization:**
   - We need to maximize the ratio \(\frac{|BC|}{|AB|}\).
   - Let \(P = \frac{BC}{AB}\).
   - To maximize \(P\), we can minimize \(\frac{1}{P}\):
     \[
     \frac{1}{P} = \frac{AB}{BC} = \frac{a - \frac{y}{k} - \sqrt{r^2 - y^2}}{2\sqrt{r^2 - y^2}}
     \]

6. **Simplification:**
   - Let \(t = \frac{(ka - y)^2}{r^2 - y^2}\).
   - We need to minimize \(t\):
     \[
     t = \frac{(ka - y)^2}{r^2 - y^2}
     \]
   - This leads to a quadratic equation:
     \[
     (1 + t)y^2 - 2kay + (ka)^2 - tr^2 = 0
     \]
   - The discriminant \(\Delta \ge 0\) implies:
     \[
     t \ge \frac{(ka)^2 - r^2}{r^2}
     \]

7. **Optimal \(y\):**
   - Solving for \(y\):
     \[
     y = \frac{r^2}{ka}
     \]
   - This gives the maximum value of \(P\).

8. **Construction:**
   - Construct \(DF \perp OD\) where \(F\) is the intersection of \(DF\) with the circle \(\omega\).
   - Construct \(GF \perp OF\) where \(G\) is the intersection of \(GF\) with \(OD\).
   - Construct \(DH \perp a\) where \(H\) is the intersection of \(FD\) with \(DH\).
   - Finally, construct the line \(\ell\) parallel to \(OD\) passing through \(H\).

The final answer is \(\boxed{\frac{r^2}{ka}}\)