Find the locus of the circle passing through O, A and B.
Let the line intersect X and Y axis at (2h,0) and (0,2k) respectively.
The line is the diameter for the given circle, so the centre will be (h,k).
The radius of the circle is $c$
Then $$h^2+k^2=c^2$$
This is the locus for the centre of the circle. However, i am not able to find the locus for the entire circle itself, because the centre isn’t clearly defined.
The answer is $x^{-2}+y^{-2}=4c^{-2}$
Let C$(x,y)$ be the center of the line segment AB. Then, the coordinates of A and B are $(2x,0)$ and $(0,2y)$, respectively. Establish the relationship below with the area of the right triangle AOB,
$$Area_{AOB} - \frac 12 OA\cdot OB = \frac 12 c\cdot AB$$
or, in terms of the coordinate $x$ and $y$ for the center,
$$(2x)(2y) = c\sqrt{(2x)^2 + (2y)^2}$$
Rearrange to obtained the equation of the locus, which is shown in blue in the graph above
$$\frac1{x^2}+\frac1{y^2} = \frac4{c^2}$$