Key Concepts and Formulas

• Equation of a Circle: The general equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$ or $x^2 + y^2 + 2gx + 2fy + c = 0$, where $g = -h$, $f = -k$, and $c = h^2 + k^2 - r^2$.
• Equation of the Common Chord: If $S_1 = 0$ and $S_2 = 0$ are the equations of two circles, then the equation of their common chord is given by $S_1 - S_2 = 0$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

Step 1: Define the circles and rewrite their equations.

We are given the equation of the first circle as $C_1: x^2 + y^2 - 2(x+y) + 1 = 0$, which can be rewritten as $x^2 + y^2 - 2x - 2y + 1 = 0$.

The second circle $C_2$ has its center at $(-1, 0)$ and radius 2. Therefore, its equation is $(x+1)^2 + (y-0)^2 = 2^2$, which expands to $x^2 + 2x + 1 + y^2 = 4$, or $x^2 + y^2 + 2x - 3 = 0$.

Step 2: Find the equation of the common chord.

To find the equation of the common chord, we subtract the equation of $C_2$ from the equation of $C_1$: $(x^2 + y^2 - 2x - 2y + 1) - (x^2 + y^2 + 2x - 3) = 0$ Simplifying, we get: $-4x - 2y + 4 = 0$ Dividing by -2, we obtain the equation of the common chord: $2x + y - 2 = 0$

Step 3: Find the coordinates of point P.

The common chord intersects the y-axis at point P. On the y-axis, $x = 0$. Substituting $x = 0$ into the equation of the common chord, we get: $2(0) + y - 2 = 0$ $y = 2$ Therefore, the coordinates of point P are $(0, 2)$.

Step 4: Find the center of circle C1.

From the equation of circle $C_1: x^2 + y^2 - 2x - 2y + 1 = 0$, we can identify the center as $(1, 1)$.

Step 5: Calculate the square of the distance between P and the center of C1.

The center of $C_1$ is $(1, 1)$ and the coordinates of point P are $(0, 2)$. The distance between these two points is: $d = \sqrt{(1 - 0)^2 + (1 - 2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$ We need to find the square of the distance, which is: $d^2 = (\sqrt{2})^2 = 2$

Step 6: Identify the Correct Answer

The calculated square distance is 2. However, the provided solution states the correct answer is 4. Let's re-examine the problem for errors. There is no error.

Let's re-examine the equation of the common chord $2x + y - 2 = 0$. When $x = 0$, we have $y = 2$, so $P = (0, 2)$. The center of $C_1$ is $(1, 1)$. The square of the distance between $(0, 2)$ and $(1, 1)$ is $(1-0)^2 + (1-2)^2 = 1^2 + (-1)^2 = 1 + 1 = 2$.

The question asks for the square of the distance of P from the centre of C1. The square of the distance between P(0, 2) and the center of C1 (1, 1) is $(0-1)^2 + (2-1)^2 = 1 + 1 = 2$.

The provided answer is incorrect. However, given the correct answer is stated as 4, let's re-examine the problem statement and solution provided. The solution is correct, thus the original given answer is incorrect. The square of the distance is 2.

Common Mistakes & Tips

• Be careful with signs when rewriting the circle equations in the general form.
• Remember to subtract the equations of the circles in the correct order to obtain the common chord equation. The order only affects the sign of the equation, which doesn't change the line itself.
• Double-check the arithmetic when calculating the distance between two points.

Summary

We first found the equations of both circles. Then, we determined the equation of the common chord by subtracting the two circle equations. We found the y-intercept of the common chord, which gave us the coordinates of point P. Finally, we calculated the square of the distance between point P and the center of circle $C_1$, obtaining a result of 2.

The final answer is \boxed{2}.