Exams › JEE Advanced › Maths
A point moves such that the sum of the squares of its distances from all four sides of a unit square is equal to 3. If the locus of this point is a circle, find its radius.
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Correct answer: 1
Solution
Let unit square have vertices at (0,0), (1,0), (1,1), (0,1). For point P(x,y), distances to sides x=0, x=1, y=0, y=1 are x, 1-x, y, 1-y (assuming P inside or near square). Sum of squares: x² + (1-x)² + y² + (1-y)² = 3. 2x² - 2x + 1 + 2y² - 2y + 1 = 3. 2x² - 2x + 2y² - 2y = 1. x² - x + y² - y = 1/2. (x - 1/2)² - 1/4 + (y - 1/2)² - 1/4 = 1/2. (x - 1/2)² + (y - 1/2)² = 1. Circle of radius 1 centered at (1/2, 1/2).
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