Question

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 ______.

पर्याय

• 320

• 160

• 105

• 210

Answer 1

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 210.

Explanation:

Circle x² + y² = 16

O(0, 0), r = 4

Chord x + y = n

OB = `|0 + 0 - "n"|/sqrt(1 + 1) = (+"n")/sqrt(2)`

AC = 2AB = `2sqrt("OA"^2 - "OB"^2)`

= `2sqrt(16 - "n"^2/2)`

AC = 2AB = `2sqrt(16 - "n"^2/2)`

n = 1   AC² = `4(16 - 1^2/2)` = 62

n = 2   AC² = `4(16 - 2^2/2)` = 56

n = 3   AC² = `4(16 - 3^2/2)` = 46

n = 4   AC² = `4(16 - 4^2/2)` = 32

n = 2   AC² = `4(16 - 5^2/2)` = 14

Sum = 210