Key Concepts and Formulas

• Equation of a Circle: The general form $x^2+y^2+2gx+2fy+c=0$ has center $(-g, -f)$ and radius $R = \sqrt{g^2+f^2-c}$. The standard form is $(x-h)^2+(y-k)^2=R^2$, where $(h,k)$ is the center and $R$ is the radius.
• Distance from a Point to a Line: The perpendicular distance from a point $(x_0, y_0)$ to a line $Ax+By+C=0$ is given by $d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$.
• Properties of an Inscribed Square: The center of the square coincides with the center of the circle. If the radius of the circle is $R$, the distance from the center of the circle to any side of the inscribed square is $R/\sqrt{2}$.

Step-by-Step Solution

Step 1: Analyze the Circle Equation

We are given the equation of the circle as $x^2+y^2-10x-6y+30=0$. We want to find the center and radius of the circle by completing the square to get the equation in standard form. $(x^2 - 10x) + (y^2 - 6y) + 30 = 0$ $(x^2 - 10x + 25) + (y^2 - 6y + 9) + 30 - 25 - 9 = 0$ $(x-5)^2 + (y-3)^2 = 4$ From this standard form, we can identify the center and radius. The center of the circle is $(5,3)$ and the radius is $R = \sqrt{4} = 2$.

Step 2: Determine the Distance from the Center to a Side of the Square

Since the square is inscribed in the circle, the distance from the center of the circle to any side of the square is $d = \frac{R}{\sqrt{2}}$. Given $R=2$, the distance is: $d = \frac{2}{\sqrt{2}} = \sqrt{2}$

Step 3: Find the Equations of the Sides of the Square

We are given that one side of the square is parallel to the line $y=x+3$. Any line parallel to $y=x+3$ will have the same slope, which is $m=1$. Therefore, the equation of a side of the square can be written as $y = x+k$, or equivalently, $x-y+k=0$.

We know that the perpendicular distance from the center of the circle $(5,3)$ to this line $x-y+k=0$ must be $\sqrt{2}$. Using the distance formula $d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$: $\sqrt{2} = \frac{|(1)(5) + (-1)(3) + k|}{\sqrt{1^2 + (-1)^2}}$ $\sqrt{2} = \frac{|5 - 3 + k|}{\sqrt{2}}$ $\sqrt{2} \times \sqrt{2} = |2+k|$ $2 = |2+k|$ This absolute value equation gives two possibilities for $2+k$:

1. $2+k = 2 \Rightarrow k = 0$
2. $2+k = -2 \Rightarrow k = -4$

These two values of $k$ correspond to two parallel sides of the square, $L_1$ and $L_2$.

• For $k=0$, the equation of one side is $y=x$.
• For $k=-4$, the equation of the opposite side is $y=x-4$.

Step 4: Calculate the Vertices of the Square

The vertices of the square are the points where these lines intersect the circle. We will solve the system of equations for each line with the circle's equation $(x-5)^2 + (y-3)^2 = 4$.

Case 1: Line $L_1 \equiv y=x$ Substitute $y=x$ into the circle equation: $(x-5)^2 + (x-3)^2 = 4$ $(x^2 - 10x + 25) + (x^2 - 6x + 9) = 4$ $2x^2 - 16x + 34 = 4$ $2x^2 - 16x + 30 = 0$ Divide by 2: $x^2 - 8x + 15 = 0$ Factor the quadratic equation: $(x-3)(x-5) = 0$ This gives two possible $x$-coordinates: $x=3$ or $x=5$. Since $y=x$:

• If $x=3$, then $y=3$. So, one vertex is $V_1=(3,3)$.
• If $x=5$, then $y=5$. So, another vertex is $V_2=(5,5)$.

Case 2: Line $L_2 \equiv y=x-4$ Substitute $y=x-4$ into the circle equation: $(x-5)^2 + ((x-4)-3)^2 = 4$ $(x-5)^2 + (x-7)^2 = 4$ $(x^2 - 10x + 25) + (x^2 - 14x + 49) = 4$ $2x^2 - 24x + 74 = 4$ $2x^2 - 24x + 70 = 0$ Divide by 2: $x^2 - 12x + 35 = 0$ Factor the quadratic equation: $(x-5)(x-7) = 0$ This gives two possible $x$-coordinates: $x=5$ or $x=7$. Since $y=x-4$:

• If $x=5$, then $y=5-4=1$. So, a third vertex is $V_3=(5,1)$.
• If $x=7$, then $y=7-4=3$. So, the fourth vertex is $V_4=(7,3)$.

The four vertices of the square are $(3,3)$, $(5,5)$, $(5,1)$, and $(7,3)$.

Step 5: Compute the Required Sum

We need to find $\Sigma(x_i^2+y_i^2)$ for all four vertices $(x_i, y_i)$. Let's calculate $x_i^2+y_i^2$ for each vertex:

• For $V_1=(3,3)$: $x_1^2+y_1^2 = 3^2+3^2 = 9+9 = 18$
• For $V_2=(5,5)$: $x_2^2+y_2^2 = 5^2+5^2 = 25+25 = 50$
• For $V_3=(5,1)$: $x_3^2+y_3^2 = 5^2+1^2 = 25+1 = 26$
• For $V_4=(7,3)$: $x_4^2+y_4^2 = 7^2+3^2 = 49+9 = 58$

Now, sum these values: $\Sigma(x_i^2+y_i^2) = 18 + 50 + 26 + 58 = 152$

Common Mistakes & Tips

• Tip 1: When using the distance formula, remember the absolute value. This can lead to multiple possible lines.
• Tip 2: Always check your solutions by substituting back into the original equations.
• Tip 3: Completing the square correctly is crucial for finding the circle's center and radius.

Summary

We first determined the center and radius of the circle. Using the property that the distance from the center to a side of the inscribed square is $R/\sqrt{2}$, we found the equations of two sides. We then solved for the intersection points of the lines and the circle, yielding the vertices of the square. Finally, we calculated the sum of $x_i^2+y_i^2$ for each vertex.

The final answer is \boxed{152}, which corresponds to option (A).