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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
(a) 154 (b) 124 (c) 172 (d) 128

A

154

B

124

C

172

D

128

Text Solution

Verified by Experts

The correct Answer is:
A
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`
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