A person P moves such that projection of his distance $ D$ to a fixed point C onto a fixed line L through C is proportional to distance left after removing constant length $L$ from C. Find his path.
EDIT1
It may be of interest to note that 1/R is eccentricity and L is latus rectum for all conics in polar form. And the question presents another new definition of conics without directrix.
EDIT2:
A quick sketch on Geogebra verifies that ratio of (projected segment length on x-axis) to (focal ray length minus latus rectum segment length) equals to chosen eccentricity $\epsilon$ of three constructed conics.
$$ \dfrac{r-p}{x}= \epsilon \quad ,\frac{EE_d}{Oe} = \epsilon_{ell},\, \frac{PP_d}{Op} = \epsilon_{par},\, \frac{HH_d}{Oh} = \epsilon_{hyp}.\, $$
Without loss of generality, let be L the polar axis, the fixed point C the origin and the coordinates of that person $P=(r,\theta)$. So, $D=r$. The projection onto L is $D\cos\theta$. $D-L$ is the diminished distance. So.
$$\frac{D\cos\theta}{D-L}=R$$
With $R$ the constant of proportionality.
$$D=\frac{RL}{R-\cos\theta}=\frac{L}{1-(1/R)\cos\theta}$$
It's an ellipse or a hyperbola depending on the value of the parameters.