Let's understand the problem step by step:

We have a circle: $x^2+y^2=16$. This is a circle centered at (0,0) with radius 4.

We are given lines: $x+y=n$, where $n\in N$ (set of natural numbers).

For each line $x+y=n$, we need to find the length of the chord it intercepts on the circle (if it does intercept). Then, we need to sum the squares of these chord lengths for all such lines where the chord exists.

Step 1: Condition for a line to intersect the circle

The perpendicular distance from the center (0,0) of the circle to the line $x+y=n$ is given by:

$d=\frac{\left|n\right|}{\sqrt{1^2+1^2}}=\frac{\left|n\right|}{\sqrt{2}}$

For the line to intersect the circle (i.e., to cut a chord), the distance from the center to the line must be less than the radius:

$d<r$ ⇒ $\frac{\left|n\right|}{\sqrt{2}}<4$ ⇒ $\left|n\right|<4\sqrt{2}$

Since $4\sqrt{2}\approx 5.656$, and n is a natural number (positive integer), the possible values of n for which the line cuts a chord are: n = 1, 2, 3, 4, 5.

Note: For n=0, the line is x+y=0 which passes through the center, so it is a diameter (chord length = 8). But since n ∈ N (natural numbers, typically starting from 1), we consider n=1 to 5.

Step 2: Find the chord length for a given line x+y=n

The length of the chord intercepted by a line at a distance d from the center of a circle of radius r is given by:

$L=2\sqrt{r^2-d^2}$

So for our circle (r=4) and line (d = |n|/√2), the chord length L is:

$L=2\sqrt{4^2-{\left(\frac{n}{\sqrt{2}}\right)}^2}=2\sqrt{16-\frac{n^2}{2}}=2\sqrt{\frac{32-n^2}{2}}=\sqrt{2}\sqrt{32-n^2}$

Then, the square of the chord length is:

$L^2=2(32-n^2)=64-2n^2$

Step 3: Sum the squares of chord lengths for n=1 to 5

We need to compute S = Σ (L²) for n=1 to 5 = Σ (64 - 2n²) from n=1 to 5

So, S = [64*5] - 2 * Σ n² from n=1 to 5

We know Σ n² from 1 to k = k(k+1)(2k+1)/6

For k=5: Σ n² = (5*6*11)/6 = 55

So, S = 320 - 2*55 = 320 - 110 = 210

Final Answer: 210

Related Topics and Formulae

Circle Geometry: The equation of a circle centered at (h,k) with radius r is (x-h)² + (y-k)² = r². The length of a chord at a distance d from the center is 2√(r²-d²).

Distance from a point to a line: The distance from point (x₁,y₁) to line ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²).

Sum of squares: The sum of squares of first n natural numbers is n(n+1)(2n+1)/6.