Let $D$ be a fixed line in $\mathbb{R^2}$. For each line $r$ non parallel to $D$ and passing through the origin $(0,0)$, let $M$ and $N$ be points on $r$ such that: $$d(M, P) = d(N, P) = d(P, A)$$ where $P = D \bigcap r$ and $A$ is the orthogonal projection of $P$ on the $x$ axis, as shown below. The set of all points $M$ and $N$ when $r$ varies is called the logocyclical or strophoid curve. Determine a parametrization for this curve.
A hint given in the book is that one should show that the polar equation of the strophoid (where $a = d(O, A) $) is $r \cos(\theta) = a(1 \pm \sin(\theta))$, but I haven't been able to do so. I tried using simple trigonometry and all the usual stuff but it was still out of my reach. I'd be very grateful for any help/hints.
A little progress: I have managed to show that if $D$ is the line $x = 1$, a parametrization for the curve is $$\alpha(t) = \left(\frac{2t^2}{t^2 + 1}, t \sqrt{1 - \frac{4t^2}{(t^2 + 1)^2}} \right)$$
but the more general case still eludes me.
$$ AK=AH=ML=AI=OA\sin\theta=a\sin\theta $$ whence: $$ OK=r_N\cos\theta=OA+AK=a+a\sin\theta, \quad OH=r_M\cos\theta=OA-AH=a-a\sin\theta. $$