Question

The sides of a square are parallel to coordinate axes and it is inscribed in the circle $x^2+y^2-2x+4y-93=0$. Find the coordinates of the vertices of the square.

Answer 1

The given circle is $(1,-2)$ and $7\sqrt{2}$. The sides are parallel to axes. Diagonal $AC$ will be making an angle of ${45}^o$ and $BD$ will make an angle ${135}^o$ with $x-axis$.
$A$ and $C$ will lie on $\frac{x-1}{\cos \theta }=\frac{y+2}{\sin \theta }=\begin{array}{c}7\sqrt{2}\\ C\end{array},\begin{array}{c}-7\sqrt{2}\\ A\end{array}$
Put $\theta ={45}^o$ $\therefore C$ and $A$ are $(8,5)$ and $(-6,-9)$
$B$ and $D$ will lie on $\frac{x-1}{\cos \theta }=\frac{y+2}{\sin \theta }=\begin{array}{c}7\sqrt{2}\\ B\end{array},\begin{array}{c}-7\sqrt{2}\\ D\end{array}$
Put $\theta ={135}^o$ $\therefore B$ and $D$ are $(-6,5)$ and $(8,-9)$