If a constant angle $\gamma$ is subtended between straight lines drawn from two fixed foci then loci are Circular Arcs. And if the product of these line segment $d_1d_2$ is a constant, then loci are Cassinian Ovals.
What are the loci if product $ (\sqrt{d_1d_2} \cos\frac{ \gamma }{2} )$ is constant ?
We have $$k = \sqrt{d_1d_2}\cos\frac{\gamma}{2}\quad\to\quad k^2 = d_1 d_2\cos^2\frac{\gamma}{2} = d_1 d_2\cdot\frac12(1+\cos\gamma) \tag{1}$$ so that, with a soon-to-be-convenient extra factor of $2$, $$4k^2 = 2d_1d_2(1+\cos\gamma) \tag{2}$$ Writing $2d$ for the distance between the foci, the Law of Cosines tells us $$(2d)^2 = d_1^2+d_2^2-2d_1d_2\cos\gamma \tag{3}$$ Combining $(2)$ and $(3)$, $$4d^2 + 4 k^2 = d_1^2 + d_2^2 + 2 d_1 d_2 = ( d_1+d_2)^2 \tag{4}$$ and we deduce $$d_1+d_2= 2 \sqrt{d^2+k^2} \tag{5}$$
The locus is (an arc of) an ellipse. $\square$
