Key Concepts and Formulas

• Distance Formula: The square of the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $D^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$.
• General Equation of a Circle: A circle can be represented by the equation $x^2 + y^2 + 2gx + 2fy + c = 0$.
• Radius of a Circle: For a circle with the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, its radius $R$ is given by the formula $R = \sqrt{g^2 + f^2 - c}$. The diameter $d$ is simply $2R$.

Step-by-Step Solution

1. Define the Moving Point and Fixed Points: Let the moving point be $P(x, y)$. The four fixed points are $A(0, 0)$, $B(1, 0)$, $C(0, 1)$, and $D(1, 1)$. We are given that the sum of the squares of the distances from $P$ to these four points is 18. This can be written as: $(PA)^2 + (PB)^2 + (PC)^2 + (PD)^2 = 18$

2. Apply the Distance Formula: We will now express each squared distance in terms of $x$ and $y$ using the distance formula:

   • $(PA)^2 = (x - 0)^2 + (y - 0)^2 = x^2 + y^2$
   • $(PB)^2 = (x - 1)^2 + (y - 0)^2 = (x - 1)^2 + y^2$
   • $(PC)^2 = (x - 0)^2 + (y - 1)^2 = x^2 + (y - 1)^2$
   • $(PD)^2 = (x - 1)^2 + (y - 1)^2$

3. Substitute and Expand: Substitute the expressions from Step 2 into the given equation: $(x^2 + y^2) + ((x - 1)^2 + y^2) + (x^2 + (y - 1)^2) + ((x - 1)^2 + (y - 1)^2) = 18$ Expand the squared terms: $x^2 + y^2 + (x^2 - 2x + 1 + y^2) + (x^2 + y^2 - 2y + 1) + (x^2 - 2x + 1 + y^2 - 2y + 1) = 18$

4. Simplify the Equation: Combine like terms: $(x^2 + x^2 + x^2 + x^2) + (y^2 + y^2 + y^2 + y^2) + (-2x - 2x) + (-2y - 2y) + (1 + 1 + 1 + 1) = 18$ $4x^2 + 4y^2 - 4x - 4y + 4 = 18$

5. Rearrange to the Standard Circle Equation: Move the constant term to the right side: $4x^2 + 4y^2 - 4x - 4y = 18 - 4$ $4x^2 + 4y^2 - 4x - 4y = 14$ Divide by 4 to get the standard form: $x^2 + y^2 - x - y = \frac{14}{4} = \frac{7}{2}$ $x^2 + y^2 - x - y - \frac{7}{2} = 0$

6. Find the Radius: Compare the equation $x^2 + y^2 - x - y - \frac{7}{2} = 0$ with the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$. We have:

   • $2g = -1 \implies g = -\frac{1}{2}$
   • $2f = -1 \implies f = -\frac{1}{2}$
   • $c = -\frac{7}{2}$ The radius $R$ is given by: $R = \sqrt{g^2 + f^2 - c} = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 - \left(-\frac{7}{2}\right)}$ $R = \sqrt{\frac{1}{4} + \frac{1}{4} + \frac{7}{2}} = \sqrt{\frac{1}{4} + \frac{1}{4} + \frac{14}{4}} = \sqrt{\frac{16}{4}} = \sqrt{4} = 2$

7. Calculate the Diameter and its Square: The diameter $d$ is $2R = 2(2) = 4$. Therefore, $d^2 = 4^2 = 16$.

Common Mistakes & Tips

• Be careful with the signs when using the distance formula and expanding the squared terms. A small mistake can lead to a completely different equation.
• Remember to divide by the coefficient of $x^2$ and $y^2$ to obtain the standard form of the circle's equation before calculating the radius.
• Double-check your arithmetic when simplifying fractions and calculating the radius.

Summary

We started by defining the moving point and the fixed points. We then used the distance formula to express the sum of the squares of the distances as an equation in terms of $x$ and $y$. After expanding and simplifying, we obtained the equation of a circle. By comparing this equation with the general form of a circle, we found the radius and hence the diameter. Finally, we calculated the square of the diameter, which is 16.

The final answer is \boxed{16}.