Key Concepts and Formulas

• Equation of a Circle: The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$. A circle centered at the origin has the equation $x^2 + y^2 = r^2$.
• Perpendicular Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
• Length of a Chord: If $r$ is the radius of a circle and $d$ is the perpendicular distance from the center of the circle to a chord, then the length $L$ of the chord is given by $L = 2\sqrt{r^2 - d^2}$, and therefore $L^2 = 4(r^2 - d^2)$.

Step-by-Step Solution

Step 1: Identify Circle Properties

The equation of the circle is $x^2 + y^2 = 16$. Comparing this with the standard form $x^2 + y^2 = r^2$, we can identify the center and radius.

• Center: $(0, 0)$
• Radius: $r = \sqrt{16} = 4$

Step 2: Identify Line Equation and Parameters

The equation of the line is $x + y = n$, which can be rewritten as $x + y - n = 0$. Comparing this with the general form $Ax + By + C = 0$, we have $A = 1$, $B = 1$, and $C = -n$. We are given that $n \in \mathbb{N}$.

Step 3: Calculate the Perpendicular Distance from the Center to the Line

We need to find the perpendicular distance $d$ from the center $(0, 0)$ to the line $x + y - n = 0$. Using the formula for the perpendicular distance: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} = \frac{|1(0) + 1(0) - n|}{\sqrt{1^2 + 1^2}} = \frac{|-n|}{\sqrt{2}} = \frac{n}{\sqrt{2}}$ Since $n$ is a natural number, $n > 0$, so $|-n| = n$.

Step 4: Determine the Condition for a Chord to Exist and Find Possible Values of n

For the line to intersect the circle and form a chord, the perpendicular distance $d$ must be less than the radius $r$: $d < r$ $\frac{n}{\sqrt{2}} < 4$ $n < 4\sqrt{2}$ Since $\sqrt{2} \approx 1.414$, $n < 4 \times 1.414 \approx 5.656$ Since $n \in \mathbb{N}$, the possible values for $n$ are $1, 2, 3, 4, 5$.

Step 5: Express the Square of the Length of the Chord ($L^2$) in terms of n

The square of the length of the chord is given by: $L^2 = 4(r^2 - d^2)$ Substitute $r = 4$ and $d = \frac{n}{\sqrt{2}}$: $L^2 = 4\left(4^2 - \left(\frac{n}{\sqrt{2}}\right)^2\right) = 4\left(16 - \frac{n^2}{2}\right) = 64 - 2n^2$

Step 6: Calculate the Sum of the Squares of the Lengths of the Chords

We need to find the sum of $L^2$ for all possible values of $n$ (i.e., $n = 1, 2, 3, 4, 5$): $\sum L^2 = \sum_{n=1}^{5} (64 - 2n^2) = \sum_{n=1}^{5} 64 - 2 \sum_{n=1}^{5} n^2$ $= 64(5) - 2(1^2 + 2^2 + 3^2 + 4^2 + 5^2) = 320 - 2(1 + 4 + 9 + 16 + 25) = 320 - 2(55) = 320 - 110 = 210$

Common Mistakes & Tips

• Incorrect Distance Formula: Ensure you use the correct formula for the perpendicular distance from a point to a line.
• Forgetting the Chord Condition: Remember that $d < r$ for a chord to exist. This limits the possible values of $n$.
• Arithmetic Errors: Be careful with calculations, especially when squaring and summing.

Summary

We found the radius of the circle and used the perpendicular distance formula to find the distance from the center to the line. We then used the condition $d < r$ to determine the possible values of $n$. Finally, we calculated the sum of the squares of the lengths of the chords for these values of $n$, arriving at the final answer.

Final Answer

The final answer is \boxed{210}, which corresponds to option (A).