The question you shared involves a square $ABCD$ on a Cartesian coordinate plane with vertices at:

• $A(-1, 1)$
• $B(1, 1)$
• $C(1, -1)$
• $D(-1, -1)$

A circle $L$ passes through these four points, and initially, this circle is rotated by 15° clockwise around point $A$. The question asks for the amount $p$ (in horizontal shift) and $q$ (in vertical shift) needed to return the circle to its original position after this rotation.

Approach to Solve:

1. Circle Details: Since the circle passes through all four vertices of square $ABCD$, the center of the circle is at the origin $(0, 0)$, and the radius is the distance from the origin to any of the vertices (like $A$), which is 1 unit (from Pythagoras' theorem).

2. Rotation: After the circle is rotated by 15° around point $A$, the goal is to shift it back to its original position by a certain amount horizontally $p$ and vertically $q$. The correct $p$ and $q$ depend on the angle of rotation and the distance of point $A$ from the origin.

3. Shifting Back: By trigonometry, the values of $p$ and $q$ correspond to the horizontal and vertical displacements after a 15° rotation. These displacements can be calculated using the sine and cosine of 15°.

4. Final Result: After calculating, the value of $p + q$ corresponds to one of the provided answer choices.

Thus, the correct answer is:

$p + q = \frac{\sqrt{6} - \sqrt{2}}{2}$

Do you need any more detailed steps or explanations on this? Also, here are 5 questions to consider that are related to this problem:

1. How do we determine the radius of the circle passing through the vertices of the square?
2. What is the significance of rotating the circle by 15° around point $A$?
3. How can trigonometric functions help calculate the shift in position after rotation?
4. Can the shifting be visualized geometrically on the coordinate plane?
5. How does the rotation affect the coordinates of the vertices?

Tip: For geometric problems involving rotation, trigonometry (sine and cosine) is often the best tool to calculate shifts and displacements!