Multiple choice

The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers, is

  1. 320

  2. 160

  3. 105

  4. 210
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D Correct answer
Explanation

The chord length L of x^2 + y^2 = r^2 cut by x + y = n is 2 * sqrt(r^2 - d^2), where d is the distance from origin to line. Here r=4, d=n/sqrt(2). L^2 = 4(16 - n^2/2) = 64 - 2n^2. Summing for n=1 to 5 (since n^2 < 32): (64-2) + (64-8) + (64-18) + (64-32) + (64-50) = 62 + 56 + 46 + 32 + 14 = 210.

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