I tried to find locus of all points such that the product of distances from two focii is constant. I assumed that the vertex is at (+a,0), and the two focii are (+c,0) and (-c,0). I arrived at the following,
$ (\frac{x^2}{a^2} + \frac{y^2}{a^2} -1)a^2 + 2(\frac{x^2}{a^2} - \frac{y^2}{a^2} +1)c^2 = 0 $
Is it correct. And Is it correct to find such locus? How do these look when plotted?
We let a point $P$ a point of coords $(x,y)$. We have to have: $$((x-c)^2+y^2)((x+c)^2+y^2)=k, k\in R_0^+$$Substituing for $c$ we have: $$((x-4)^2+y^2)((x+4)^2+y^2)=k$$ This locus changes in function of $k$.