All chords through an external point to the circle `x^2+y^2= 16` are drawn having length `l` which is a positive integer. The sum of the squares of the distances from centre of circle to these chords is

Options

A. 154
B. 124
C. 172
D. 128

Answer

A

Explanation

Chords are of lengths, `l =1,2,3,4,5,6,7,8,7,6,5,4,3,2,1`
`:.` Total number of chords `= 15`
Length of chord `= 2 sqrt(r^(2)-d^(2))` (where r is radius and d is distance of chord from center).
`:. 4(Sigma r^(2) -Sigma d^(2)) = 2(1^(2) + 2^(2)+...+7^(2)) +8^(2)`
`rArr 4(Sigma r^(2) - Sigma d^(2)) = (2.(7)(8)(15))/(6) +8^(2)`
`rArr Sigma d^(2) = Sigma r^(2) -(344)/(4)`
`rArr Sigma d^(2) = 15 (16)-86`
`rArr Sigma d^(2) = 154`