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Step-by-Step Solution
Step 1: Identify the known points and the variable point
Let the four fixed points be A(0, 0), B(1, 0), C(0, 1), and D(1, 1). Let the moving point be P(x, y) whose locus we need to determine.
Step 2: Express the sum of squares of distances
We are given that the sum of the squares of the distances from P(x, y) to each of A, B, C, and D is 18:
(PA)² + (PB)² + (PC)² + (PD)² = 18.
In coordinate form, each distance squared is written as follows:
(PA)² = x² + y²
(PB)² = (x − 1)² + y²
(PC)² = x² + (y − 1)²
(PD)² = (x − 1)² + (y − 1)²
Step 3: Substitute and simplify the equation
Summing these up and setting equal to 18:
x^2 + y^2 + (x - 1)^2 + y^2 + x^2 + (y - 1)^2 + (x - 1)^2 + (y - 1)^2 = 18.
Expand the squares and combine like terms carefully:
x^2 + (x - 1)^2 = x^2 + (x^2 - 2x + 1) = 2x^2 - 2x + 1.
y^2 + (y - 1)^2 = y^2 + (y^2 - 2y + 1) = 2y^2 - 2y + 1.
Each pair appears twice, so total:
2(2x^2 - 2x + 1) + 2(2y^2 - 2y + 1).
This simplifies to:
4x^2 - 4x + 2 + 4y^2 - 4y + 2 = 18,
4x^2 + 4y^2 - 4x - 4y + 4 = 18.
4x^2 + 4y^2 - 4x - 4y = 14.
Divide through by 4 to simplify:
x^2 + y^2 - x - y = \frac{14}{4} = \frac{7}{2}.
Hence, we get the equation of the locus as:
x^2 + y^2 - x - y - \frac{7}{2} = 0.
Step 4: Complete the square to find the center and radius
Rewrite the equation by completing the square for both x and y:
x^2 - x + \frac{1}{4} + y^2 - y + \frac{1}{4} = \frac{7}{2} + \frac{1}{4} + \frac{1}{4}.
(x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = \frac{7}{2} + \frac{1}{2}.
(x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = 4.
This represents a circle with center at \left(\frac{1}{2}, \frac{1}{2}\right) and radius r = 2. Thus, the diameter d = 2r = 4.
Step 5: Conclude the required value of d^2
We have d = 4, so d^2 = 16.
Final Answer
The value of d^2 is 16.
