A square is inclined in the circle `x^2 + y^2 - 10 x - 6y + 30 = 0` such that one side of the 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

Options

A. 148
B. 156
C. 152
D. 160

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the Circle's Center and Radius
The given equation of the circle is:
\[ x^2 + y^2 - 10x - 6y + 30 = 0 \]

We can rewrite this in standard form by completing the square.

1. Rearranging the equation:
\[ (x^2 - 10x) + (y^2 - 6y) + 30 = 0 \]

2. Completing the square for \(x\):
\[ x^2 - 10x = (x - 5)^2 - 25 \]

3. Completing the square for \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

4. Substitute back into the equation:
\[ (x - 5)^2 - 25 + (y - 3)^2 - 9 + 30 = 0 \]
\[ (x - 5)^2 + (y - 3)^2 - 4 = 0 \]
\[ (x - 5)^2 + (y - 3)^2 = 4 \]

From this, we see that the center of the circle is \((5, 3)\) and the radius \(r = 2\).

### Step 2: Determine the Orientation of the Square
The problem states that one side of the square is parallel to the line \(y = x + 3\). The slope of this line is 1, which means the sides of the square will have slopes of 1 and -1.

### Step 3: Calculate the Square's Vertices
Since the square is inscribed in the circle, the diagonals of the square will intersect at the center of the circle \((5, 3)\) and will be perpendicular to each other.

1. The diagonal of the square will be parallel to the x-axis and y-axis, respectively, since the slopes of the sides are 1 and -1.

2. The length of the diagonal of the square is equal to the diameter of the circle, which is \(2r = 4\).

3. The distance from the center to a vertex along the diagonal (half the diagonal) is:
\[ \frac{4}{2} = 2 \]

4. Thus, the vertices of the square can be calculated by moving 2 units in the directions defined by the slopes:
- Moving along the x-axis:
- Right: \((5 + 2, 3) = (7, 3)\)
- Left: \((5 - 2, 3) = (3, 3)\)
- Moving along the y-axis:
- Up: \((5, 3 + 2) = (5, 5)\)
- Down: \((5, 3 - 2) = (5, 1)\)

The vertices of the square are:
- \((7, 3)\)
- \((3, 3)\)
- \((5, 5)\)
- \((5, 1)\)

### Step 4: Calculate the Required Sum
We need to find the sum of \(x_i^2 + y_i^2\) for the vertices:
1. For vertex \((7, 3)\):
\[ 7^2 + 3^2 = 49 + 9 = 58 \]
2. For vertex \((3, 3)\):
\[ 3^2 + 3^2 = 9 + 9 = 18 \]
3. For vertex \((5, 5)\):
\[ 5^2 + 5^2 = 25 + 25 = 50 \]
4. For vertex \((5, 1)\):
\[ 5^2 + 1^2 = 25 + 1 = 26 \]

Now, summing these values:
\[ 58 + 18 + 50 + 26 = 152 \]

### Final Answer
Thus, the value of \(\sum (x_i^2 + y_i^2)\) is \(152\).

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