The sum of the squares of the lengths of the chords intercepted on the circle, x 2 + y 2 = 16 , by the lines, x + y = n , n ∈ N , where N is the set of all natural numbers is:

Step-by-step solution

The length of the perpendicular from centre 0 , 0 to the line x + y = n is 0 + 0 - n 1 2 + I 2 = n 2 Length of intercept = 2 4 2 - n 2 2 = 64 - 2 n 2 ⇒ Possible values of n are 1 , 2 , … . 5 ⇒ Sum of the square of lengths of intercepts = Σ n = 1 5 64 - 2 n 2 = 64 × 5 - 2 Σ n = 1 5 n 2 = 320 - 2 Σ n = 1 5 n 2 = 320 - 110 = 210