Locus of a moving point when tan of half look angle is inversely proportional to distance.

Question

A point $ P$ moves such that the tan of bisected angle $\alpha $ subtended between two fixed points $F_1$ and $F_2$ at $P$ is inversely proportional to distance $d$ of $P$ to line $F_1-F_2$.

Show that the locus of P is a conic.

Answer 1

Consider $\triangle FGH$, where $F$ and $G$ are the fixed points, and such that variable point $H$ is constrained by $$d\tan\theta=\lambda \tag1$$ where $\theta:=\frac12\angle FHG$, $d$ is the altitude from $H$, and $\lambda$ is some constant of proportionality. Consider also the incircle of $\triangle FGH$ noting the easily-derived fact that the tangent segment to $H$ has length $(f+g-h)/2$ (where the sides have been named in the standard way), so that we have $$\text{inradius} = \frac12(f+g-h)\tan\theta \tag2$$

Calculating the area of $\triangle FGH$ in three(!) ways (Heron's formula, "one-half base-times-height", and "one-half perimeter-times-inradius"), we can express the square of the area in two ways:

$$\begin{align} \frac1{16}(f+g+h)(-f+g+h)(f-g+h)(f+g-h)&=\frac12hd\cdot\frac14(f+g+h)(f+g-h)\tan\theta \tag3 \\[1em] \to\qquad (-f+g+h)(f-g+h) &= 2hd\tan\theta \tag4 \end{align}$$ Invoking $(1)$, we can write

$$h^2-(f-g)^2=2h\lambda \quad\to\quad |f-g|=\sqrt{h(h-2\lambda)} \tag{$\star$}$$

As the (absolute) difference of the distances from $H$ to $F$ and $G$ is constant, $H$ traces a hyperbola. $\square$

(The reader is invited to confirm a fact suggested by the image: The point of tangency of $\overline{FG}$ with the incircle is a vertex of the hyperbola.)

Answer 2

Let $PF_1=x$, $PF_2=y$, $F_1F_2=2c$. Then we have by the cosine law: $$ 4c^2=x^2+y^2-2xy\cos2\alpha, \quad\text{that is}\quad x^2+y^2=4c^2+2xy\cos2\alpha. $$ On the other hand: $$ 2c\cdot d=xy\sin2\alpha, \quad\text{that is}\quad xy={2cd\over\sin2\alpha}={cd\over\sin\alpha\cos\alpha}. $$ Hence: $$ (x-y)^2=x^2+y^2-2xy=4c^2+2xy(\cos2\alpha-1)=\\ 4c^2+2{cd\over\sin\alpha\cos\alpha}(-2\sin^2\alpha)= 4c^2-4c(d\tan\alpha)=\text{constant}. $$