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The sum of the squares of the length of ...

The sum of the squares of the length of the chords intercepted on the circle `x^(2)+y^(2)=16`, by the lines x +y = n , `n in N`, where N is the set of all natural numbers, is

A

320

B

105

C

160

D

210

Text Solution

Verified by Experts

Given equation of line is `x + y = n ,n in N" "...(i)`
and equation of circle is `x^(2)+y^(2)=16" "...(ii)`
Now, for intercept, made by circle (ii) with line (i)

`rArr (n)/(sqrt2)lt 4`
[`therefore` d = perpendicular distance from (0,0) to the line
`x+y=n " and itequal to" (|0+0-n|)/(sqrt(1^(2)+1^(2)))=(n)/(sqrt2)`]
`rArr nlt 4sqrt2" "...(iii)`
Clearly, length of chord `AB =2sqrt(4^(2)-d^(2))`
`=2sqrt(16-(n^(2))/(2))" "[therefored=(n)/(sqrt2)]`
`therefore` Sum of square of all possible length ofchords (for n = 1, 2,3,4,5)
`=4[(16xx5)-(1)/(2)(1^(2)+2^(2)+3^(2)+4^(2)+5^(2))]`
`=320-2(5(6)(11))/(6)=320-110=210`
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