What is the shape of the graph $|z-1|+|z+i|=2$ in the complex plane?
What is the shape of the graph $|z-1|+|z+i|=2$ in the complex plane?
$(A)\text{two points}\hspace{1cm}(B)\text{a line}\hspace{1cm}(C)\text{a parabola}\hspace{1cm}(D)\text{an ellipse}$
Let us take $z=x+iy$
$|(x-1)+iy|+|x+i(y+1)|=2$
$\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y+1)^2}=2$
Upon simplifying,$3x^2+3y^2-4x+4y-2xy=0$
But with this equation, i cannot tell the shape of the graph.Please help me.
Hint: $|a-b|$ represents the distance between the two points a and b in the complex plane. What geometric shape is defined by the sum of two distances being constant ? :-$)$