Question

[Key Concept: A circle touching the $x$-axis at the origin has centre $(0,r)$; eliminate $r$ for the locus.]

Equation of $\odot$ is

$$\begin{matrix} & & x^{2} + (y - \lambda)^{2} = \lambda^{2} \\ & x^{2} + y^{2} - 2\lambda y = 0 & \text{(1)} \end{matrix}$$

Let the pole be $(h,k)$.It's polar w.r.t. (1) is

$$\begin{matrix} & & xh + ky - \lambda(k + y) = 0 \\ & xh + (k - \lambda)y - \lambda k = 0 & \text{(2)} \end{matrix}$$

compare it with $lx + my + n = 0$ and eliminate $\lambda$.