A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0.$$      Its sides are parallel to the coordinate axes. The one vertex of the square is-

Solution :
The given circle is $${x^2} + {y^2} - 2x + 4y + 3 = 0.$$      Centre $$\left( {1,\, - 2} \right).$$  Lines through centre $$\left( {1,\, - 2} \right)$$  and parallel to axes are $$x=1$$  and $$y=-2$$

Let the side of square be $$2k.$$
Then sides of square are $$x =1-k$$   and $$x=1+k$$
and $$y=-2-k$$   and $$y=-2+k$$
$$\therefore $$ Co-ordinates of $$P,\,Q,\,R, \,S$$    are $$\left( {1 + k,\, - 2 + k} \right),\,\left( {1 - k,\, - 2 + k} \right),\,\left( {1 - k,\, - 2 - k} \right),\,\left( {1 + k,\, - 2 - k} \right)$$             respectively.
Also $$P\left( {1 + k,\, - 2 + k} \right)$$    lies on circle
$$\eqalign{ & \therefore {\left( {1 + k} \right)^2} + {\left( { - 2 + k} \right)^2} - 2\left( {1 + k} \right) + 4\left( { - 2 + k} \right) + 3 = 0 \cr & \Rightarrow 2{k^2} = 2 \cr & \Rightarrow k = 1\,\,{\text{or}}\,\, - 1 \cr & {\text{If }}k = 1, \cr & P\left( {2,\, - 1} \right),\,Q\left( {0,\, - 1} \right),\,R\left( {0,\, - 3} \right),\,S\left( {2,\, - 3} \right) \cr & {\text{If }}k = - 1, \cr & P\left( {0,\, - 3} \right),\,Q\left( {2,\, - 3} \right),\,R\left( {2,\, - 1} \right),\,S\left( {0,\, - 1} \right) \cr} $$