Key Concepts and Formulas

• Parametric Equation of a Circle: A circle with center (h, k) and radius r can be parameterized as x = h + rcos(θ), y = k + rsin(θ).
• Distance Formula: The squared distance between two points (x1, y1) and (x2, y2) is (x2 - x1)^2 + (y2 - y1)^2.
• Trigonometric Identity: sin^2(θ) + cos^2(θ) = 1

Step-by-Step Solution

1. Parameterize Point P on the Circle

• What and Why: We need to represent a general point P on the given circle (x - 1)^2 + (y - 1)^2 = 1. Using the parametric form of a circle simplifies distance calculations.
• Math: The center of the circle is (1, 1) and the radius is 1. Therefore, any point P on the circle can be represented as: $P = (1 + \cos\theta, 1 + \sin\theta)$

2. Calculate (PA)^2

• What and Why: We want to find the squared distance between point P and point A(1, 4) using the distance formula. This is necessary to compute (PA)^2 + (PB)^2.
• Math: $A = (1, 4)$ $(PA)^2 = ((1 + \cos\theta) - 1)^2 + ((1 + \sin\theta) - 4)^2$ $(PA)^2 = (\cos\theta)^2 + (\sin\theta - 3)^2$ $(PA)^2 = \cos^2\theta + \sin^2\theta - 6\sin\theta + 9$

3. Calculate (PB)^2

• What and Why: Similar to the previous step, we calculate the squared distance between point P and point B(1, -5) using the distance formula.
• Math: $B = (1, -5)$ $(PB)^2 = ((1 + \cos\theta) - 1)^2 + ((1 + \sin\theta) - (-5))^2$ $(PB)^2 = (\cos\theta)^2 + (\sin\theta + 6)^2$ $(PB)^2 = \cos^2\theta + \sin^2\theta + 12\sin\theta + 36$

4. Calculate (PA)^2 + (PB)^2

• What and Why: We sum the squared distances calculated in the previous two steps. This is the expression we want to maximize.
• Math: $(PA)^2 + (PB)^2 = (\cos^2\theta + \sin^2\theta - 6\sin\theta + 9) + (\cos^2\theta + \sin^2\theta + 12\sin\theta + 36)$ $(PA)^2 + (PB)^2 = 1 - 6\sin\theta + 9 + 1 + 12\sin\theta + 36$ $(PA)^2 + (PB)^2 = 47 + 6\sin\theta$

5. Maximize (PA)^2 + (PB)^2

• What and Why: We want to find the maximum value of the expression 47 + 6sin(θ). Since 47 is constant, we only need to maximize sin(θ).
• Math: The maximum value of sin(θ) is 1. Therefore, the maximum value of (PA)^2 + (PB)^2 is: $(PA)^2 + (PB)^2 = 47 + 6(1) = 53$ This maximum occurs when $\sin\theta = 1$.

6. Find the Coordinates of P

• What and Why: We need to find the coordinates of point P when sin(θ) = 1. We also need to find the corresponding value of cos(θ).
• Math: If $\sin\theta = 1$, then $\theta = \frac{\pi}{2}$ and $\cos\theta = 0$. Therefore, the coordinates of P are: $P = (1 + \cos\theta, 1 + \sin\theta) = (1 + 0, 1 + 1) = (1, 2)$

7. Determine the Relationship Between P, A, and B

• What and Why: We need to determine the geometric relationship between the points P(1, 2), A(1, 4), and B(1, -5).
• Math: The points are: $P = (1, 2)$ $A = (1, 4)$ $B = (1, -5)$ Since all three points have the same x-coordinate (x = 1), they lie on the same vertical line. Therefore, the points are collinear.

Common Mistakes & Tips

• Always use the parametric form of the circle when dealing with optimization problems involving points on a circle.
• Be careful with signs when calculating the squared distances.
• Remember the range of sine and cosine functions.

Summary

By using the parametric representation of the circle, we expressed the sum of squared distances as a function of a single variable (θ). Maximizing this expression led us to find the coordinates of point P as (1, 2). Since points P, A, and B have the same x-coordinate, they lie on a straight line.

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