Question:

Step 1: Determine the center and radius of the given circle. The equation of the circle is $x^2 + y^2 - 2x + 4y - 93 = 0$. To find its center and radius, we complete the square: $$ (x^2 - 2x) + (y^2 + 4y) = 93 $$ Add and subtract the square of half the coefficients of $x$ and $y$: $$ (x^2 - 2x + 1) - 1 + (y^2 + 4y + 4) - 4 = 93 $$ Rewrite in standard form $(x-h)^2 + (y-k)^2 = R^2$: $$ (x-1)^2 + (y+2)^2 = 93 + 1 + 4 $$ $$ (x-1)^2 + (y+2)^2 = 98 $$ From this, the center of the circle is $(h,k) = (1, -2)$ and the square of the radius is $R^2 = 98$. Step 2: Determine the geometric properties of the inscribed square. Since the sides of the square are parallel to the coordinate axes, its diagonals are parallel to the lines $y=x$ and $y=-x$. Crucially, the center of the square coincides with the center of the circle, which is $(1, -2)$. Let the coordinates of the vertices of the square be $(x, y)$. The distance from the center $(1, -2)$ to any vertex $(x, y)$ is equal to the radius $R$ of the circle. For a square with sides parallel to the coordinate axes and center $(h, k)$, its vertices are of the form $(h \pm a, k \pm a)$ for some positive value $a$. The distance from the center $(h,k)$ to a vertex $(h \pm a, k \pm a)$ is given by the distance formula: $$ \sqrt{((h \pm a) - h)^2 + ((k \pm a) - k)^2} = R $$ $$ \sqrt{(\pm a)^2 + (\pm a)^2} = R $$ $$ \sqrt{a^2 + a^2} = R $$ $$ \sqrt{2a^2} = R $$ $$ a\sqrt{2} = R $$ Solving for $a$: $$ a = \frac{R}{\sqrt{2}} $$ We know $R^2 = 98$, so $R = \sqrt{98}$. Substitute this value: $$ a = \frac{\sqrt{98}}{\sqrt{2}} = \sqrt{\frac{98}{2}} = \sqrt{49} = 7 $$ Step 3: Calculate the coordinates of the vertices. The center of the square is $(h,k) = (1, -2)$ and the value $a=7$. The coordinates of the vertices are given by $(h \pm a, k \pm a)$. Substituting the values: $$ (1 \pm 7, -2 \pm 7) $$ This gives four possible combinations for the vertices: 1. Using $(1+7, -2+7)$: $(8, 5)$ 2. Using $(1+7, -2-7)$: $(8, -9)$ 3. Using $(1-7, -2+7)$: $(-6, 5)$ 4. Using $(1-7, -2-7)$: $(-6, -9)$ Step 4: Match the calculated vertices with the given options. The calculated coordinates of the vertices of the inscribed square are $(8, 5)$, $(8, -9)$, $(-6, 5)$, and $(-6, -9)$. Let's check the given options: * Option 1: $(8, 5)$ - This matches one of the calculated vertices. * Option 2: $(8, 9)$ - This does not match any of the calculated vertices. * Option 3: $(-6, 5)$ - This matches one of the calculated vertices. * Option 4: $(-6, -9)$ - This matches one of the calculated vertices. The coordinates of the vertices are $(8, 5)$, $(-6, 5)$, $(8, -9)$, and $(-6, -9)$. The final answer is $\boxed{\text{1,3,4}}$.