The sum of 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

Engineering

Mathematics

Line and Circle

Question

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

160

210

105

320

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Equation of the circle: $x^{2} + y^{2} = 16$

Equation of the chord: $x + y = n, n \in ℕ$

Length of the chord: $L = 2 \sqrt{16 - \frac{n^{2}}{2}}$

Square of the length of the chord: $L^{2} = 2 (32 - n^{2})$

Possible values of $n$ : $1, 2, 3, 4, 5$

Sum of squares of the lengths of chords:

$2 (31) + 2 (28) + 2 (23) + 2 (16) + 2 (7) = 210$