A Triangle is formed by the lines $X + Y = 0, X - Y = 0$ and $LX + MY = 1$ where $L^2 + M^2= 1$. Find the locus of circumcenter of triangle so formed. Using basic Geometry I have gotten the coordinates of the circumcenter in terms of $L$ and $M$, I don't have any idea how to proceed further. 
To obtain the coordinates, solve the three equations taking $2$ at a time. 
Since the triangle is a right triangle the circumcenter is on the midpoint of the hypotenuse.
After solving I got, $C= (L/L^2  - M^2   ,M/L^2  - M^2)$.
How do I find a locus for this point.
Thanks in advance.

Let's call $x=\frac{L}{L^2-M^2}$ and $y=\frac{M}{L^2-M^2}$. Then $$x^2-y^2=\frac1{L^2-M^2}$$and $$x^2+y^2=\frac{M^2+L^2}{(L^2-M^2)^2}=\frac{1}{(L^2-M^2)^2}=(x^2-y^2)^2$$
From here $$x^4-2x^2y^2+y^4-x^2-y^2=0$$This looks something like
