Locus of the poles $M$ is the parabola $P_{2}$.

vanstraelen wrote: Locus of the poles $M$ is the parabola $P_{2}$. Why? ._.

The tangent line $2\lambda x+y-2p\lambda^{2}=0$ in a point $T(2p\lambda,-2p\lambda^{2})$ of the parabola $x^{2}=-2py$. Two points on this tangent line, $P(0,2p\lambda^{2})$ and $Q(p\lambda,0)$. The polar lines of these points are $y+2p\lambda^{2}=0$ and $\lambda x-y=0$, cutting in the pole $M$ of the tangent line. Eliminating the parameter $\lambda$, we find $x^{2}+2py=0$.