Question 1036348
This might be a slow, inefficient way to the solution,
but I cannot think of a better one right now.
There must be a physics (electricity and magnetism) simulation program that graphs this locus nicely, but I do not see how it would help.
 
Since P and Q are different points, {{{R(x,y)}}} is a third different point.
{{{PR/QR=1/2}}} and we know that {{{PR<>0}}} , so
{{{PR/QR=1/2}}}-->{{{PR^2/QR^1=1/4}}}<-->{{{4PR^2=QR^2}}}
Now we get to use {{{x}}} and {{{y}}} to find the equation representing the locus of point {{{R(x,y)}}} .
{{{PR^2=(x-2)^2+y^2}}}
{{{QR^2=x^2+(y+2)^2}}}
{{{4((x-2)^2+y^2)=x^2+(y+2)^2}}}
{{{4(x^2-4x+4+y^2)=x^2+y^2+4y+4}}}
{{{4x^2-16x+16+4y^2=x^2+y^2+4y+4}}}
{{{3x^2-16x+3y^2-4y=4-16}}}
{{{3x^2-16x+3y^2-4y=-12}}}
{{{x^2-(16/3)x+y^2-(4/3)y=-4}}}
{{{x^2-(16/3)x+(8/3)^2+y^2-(4/3)y+(2/3)^2=-4+(8/3)^2+(2/3)^2}}}
{{{(x-8/3)^2+(y-2/3)^2=-36/9+64/9+4/9}}}
{{{(x-8/3)^2+(y-2/3)^2=32/9}}}
That is the equation of a circle,
centered at {{{C(8/3,2/3)}}} , 
with radius {{{sqrt(32/9)=4sqrt(2)/3}}} .