Quadratic roots condition and AM-GM minimization

Expanding the equation: $x^2 - 2x + 1 + 2kx + 11 = 0 \implies x^2 + 2(k-1)x + 12 = 0$

For this equation to have no real roots, its discriminant $D < 0$: $4(k-1)^2 - 48 < 0 \implies (k-1)^2 < 12$

Since $\sqrt{12} \approx 3.46$, we have: $-3.46 < k - 1 < 3.46 \implies -2.46 < k < 4.46$

The largest integer $k$ satisfying this inequality is $k = 4$.

Now, for $y > 0$, we minimize $\frac{4}{4y} + 9y = \frac{1}{y} + 9y$.

By AM-GM inequality: $\frac{1}{y} + 9y \ge 2 \sqrt{\frac{1}{y} \cdot 9y} = 2 \times 3 = 6$

Thus, the least possible value is $6$.