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:

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TestHub · Accepted Answer

The length of the perpendicular from centre $(0, 0)$ to the line $x + y = n$ is $|\frac{0 + 0 - n}{\sqrt{1^{2} + I^{2}}}| = \frac{n}{\sqrt{2}}$

Length of intercept $= 2 \sqrt{4^{2} - {(\frac{n}{\sqrt{2}})}^{2}} = \sqrt{64 - 2 n^{2}}$

$\Rightarrow$ Possible values of $n$ are $1, 2, \dots . 5$

$\Rightarrow$ Sum of the square of lengths of intercepts

$= Σ_{n = 1}^{5} (64 - 2 n^{2}) = 64 \times 5 - 2 Σ_{n = 1}^{5} n^{2}$

$= 320 - 2 Σ_{n = 1}^{5} n^{2} = 320 - 110 = 210$

Mathematics - Circle Question with Solution | TestHub

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