Question

A square is inscribed in the circle \( x^2 + y^2 - 10x - 6y + 30 = 0 \). One side of this square is parallel to \( y = x + 3 \). If \( (x_i, y_i) \) are the vertices of the square, then \( \sum (x_i^2 + y_i^2) \) is equal to:

• A. 148
• B. 156
• C. 160
• D. 152

Answer 1

The Correct Option is D

Circle Equation Rewritten:
The initial equation for the circle is: \[ x^2 + y^2 - 10x - 6y + 30 = 0 \]
To identify the center and radius, the equation is rewritten by completing the square for both \(x\) and \(y\): \[ (x^2 - 10x) + (y^2 - 6y) = -30 \]
After completing the square:
\[ (x - 5)^2 - 25 + (y - 3)^2 - 9 = -30 \]
\[ (x - 5)^2 + (y - 3)^2 = 4 \]
Consequently, the circle's center is \((5, 3)\) and its radius is \(2\). 

Properties of the Inscribed Square: 
As the square is inscribed within the circle, its diagonal is equivalent to the circle's diameter. 
The circle's diameter is \(2 \times 2 = 4\), thus the square's diagonal measures \(4\). 

Square Side Length Calculation: 
The side length \(s\) of a square with diagonal length \(d\) is determined by the formula:
\[ s = \frac{d}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

Determination of Square Vertices: 
Given that one side of the square is parallel to the line \(y = x + 3\), the square is oriented at a 45-degree angle. The center of the square coincides with the center of the circle, which is \((5, 3)\). 
 

Using the center \((5, 3)\) and the calculated side length of \(2\sqrt{2}\), the vertices of the square are found to be:
\[ \left(5 + \frac{2}{\sqrt{2}}, 3 + \frac{2}{\sqrt{2}}\right), \quad \left(5 - \frac{2}{\sqrt{2}}, 3 + \frac{2}{\sqrt{2}}\right), \]
\[ \left(5 + \frac{2}{\sqrt{2}}, 3 - \frac{2}{\sqrt{2}}\right), \quad \left(5 - \frac{2}{\sqrt{2}}, 3 - \frac{2}{\sqrt{2}}\right) \] 
 

Calculation of \(\sum(x_i^2 + y_i^2)\): 
For each vertex \((x_i, y_i)\) of the square, the following relationship holds:
\[ x_i^2 + y_i^2 = \left(5 \pm \frac{2}{\sqrt{2}}\right)^2 + \left(3 \pm \frac{2}{\sqrt{2}}\right)^2 \]
Upon simplification for each vertex and subsequent summation, the result is: \[ \sum(x_i^2 + y_i^2) = 152 \]