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Step-by-Step Solution
Step 1: Understand the Problem
We are given the circle
$ x^2 + y^2 = 16 $
and lines of the form
$ x + y = n $,
where
$ n $ is a natural number. We need to find the sum of the squares of the lengths of the chords that these lines intercept on the given circle.
Step 2: Identify the Center and Radius of the Circle
The given circle
$ x^2 + y^2 = 16 $
is centered at the origin
$ O(0, 0) $
and has radius
$ r = 4 $.
Step 3: Express the Distance of the Line from the Center
Consider the chord formed by the line
$ x + y = n $.
The perpendicular distance from the center
$ O(0, 0) $
to the line
$ x + y = n $
is calculated as:
$ OM = \left| \frac{0 + 0 - n}{\sqrt{1^2 + 1^2}} \right| = \left| \frac{-n}{\sqrt{2}} \right| = \frac{n}{\sqrt{2}}. $
Step 4: Determine Possible Values of n
For a chord to exist, the distance from the center to the line must be less than the radius:
$ \frac{n}{\sqrt{2}} < 4 \quad \Rightarrow \quad n < 4\sqrt{2} \approx 5.65.
$
Since
$ n $
is a natural number,
$ n $
can be
$ 1, 2, 3, 4, 5. $
Step 5: Find the Length of the Chord
Using the right triangle formed by the radius, the perpendicular from the center, and half the chord, the length of the chord
$ AB $
is:
$ AB = 2 \sqrt{r^2 - \left(\frac{n}{\sqrt{2}}\right)^2}
= 2 \sqrt{16 - \frac{n^2}{2}}
= \sqrt{64 - 2n^2}.
$
Therefore, the square of the chord length is
$ \left( \sqrt{64 - 2n^2} \right)^2 = 64 - 2n^2. $
Step 6: Compute Each Chord’s Square of Length and Sum Them Up
We list the possible values of
$ n $
and compute
$ l^2 = (AB)^2 = 64 - 2n^2 $
for each:
For n = 1:
$ l^2 = 64 - 2\times (1^2) = 64 - 2 = 62. $
For n = 2:
$ l^2 = 64 - 2\times (2^2) = 64 - 8 = 56. $
For n = 3:
$ l^2 = 64 - 2\times (3^2) = 64 - 18 = 46. $
For n = 4:
$ l^2 = 64 - 2\times (4^2) = 64 - 32 = 32. $
For n = 5:
$ l^2 = 64 - 2\times (5^2) = 64 - 50 = 14. $
The sum of these squares is:
$ 62 + 56 + 46 + 32 + 14 = 210.
$
Step 7: Conclude the Correct Answer
Hence, the sum of the squares of the lengths of all such chords is
$ 210. $
Therefore, the correct answer is
210.
