<s>[INST] If there exist real numbers $a,b,c,d$ for which $f(a),f(b),f(c),f(d)$ form a square on the complex plane.Find the area of the square. For all real numbers $x$,let the mapping $f(x)=\frac{1}{x-i},\text{where} i=\sqrt{-1}$.If there exist real numbers $a,b,c,d$ for which $f(a),f(b),f(c),f(d)$ form a square on the complex plane.Find the area of the square.

$f(a)=\frac{1}{a-i}=\frac{a+i}{a^2+1}=(\frac{a}{a^2+1},\frac{1}{a^2+1})$
Similarly,$f(b)=\frac{1}{b-i}=\frac{b+i}{b^2+1}=(\frac{b}{b^2+1},\frac{1}{b^2+1})$
Similarly,$f(c)=\frac{1}{c-i}=\frac{c+i}{c^2+1}=(\frac{c}{c^2+1},\frac{1}{c^2+1})$
Similarly,$f(d)=\frac{1}{d-i}=\frac{d+i}{d^2+1}=(\frac{d}{d^2+1},\frac{1}{d^2+1})$
Now the area of the square$=(\frac{a}{a^2+1}-\frac{b}{b^2+1})^2+(\frac{1}{a^2+1}-\frac{1}{b^2+1})^2$
But this expression simplifies to $\frac{(a-b)^2}{(1+a^2)(1+b^2)}$.
How should i prove the area of the square to be $\frac{1}{2}$.Is my approach not correct?Please help me.Thanks.
 [/INST] Hint: The function $f$ bijectively maps the extended real line into the circle through the origin having centre at $\frac{i}{2}$. The area of a square inscribed in a circle with radius $r$ is $2r^2$.
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