The sum of the squares of the lengths of the chords intercepted on the circle, x² + y² = 16, by the

Q: Wait, why do we reject the negative root in the distance formula?

The distance formula $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ uses an absolute value because distance must always be a non-negative quantity. In our case, the distance from the origin to the line $x + y - n = 0$ is $\frac{|-n|}{\sqrt{2}}$. Since $n$ is a natural number ($n > 0$), $|-n|$ is simply $n$. So, the distance is $\frac{n}{\sqrt{2}}$. It's not about rejecting a root; it's about ensuring the physical distance is positive!

Q: Why is $n$ limited to $\{1, 2, 3, 4, 5\}$?

Great question! A line only forms a chord if it actually cuts through the circle. This happens only when the perpendicular distance $d$ from the center to the line is less than the radius $r$. We found $d = \frac{n}{\sqrt{2}}$ and $r = 4$. So, we need $\frac{n}{\sqrt{2}} < 4$, which means $n < 4\sqrt{2} \approx 5.65$. Since $n$ must be a natural number, $n$ can only be $1, 2, 3, 4,$ or $5$. If $n=6$, the line is outside the circle!

Q: Why can't we just use energy conservation or some other method?

Energy conservation is for dynamics! Here, we are dealing with static geometry. The problem is purely about the intersection of a circle and a line. The most direct and elegant way to solve this is by using the geometric relationship between the radius, the distance to the chord, and the chord length itself. It's a classic right-triangle problem!

Question

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n, n \in N$, where $N$ is the set of all natural numbers, is :

Options:

(A) $320$

(B) $160$

(C) $105$

(D) $210$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the origin of a coordinate plane. Around you, a perfect circle is drawn with the equation $x^2 + y^2 = 16$.

This tells us immediately that the center is at $(0,0)$ and the radius $r$ is $\sqrt{16} = 4$.

Now, consider a family of lines, $x + y = n$, where $n$ is a natural number. These lines are all parallel, each with a slope of $-1$, marching steadily away from the origin as $n$ increases. Our goal is to find the sum of the squares of the lengths of the chords these lines carve out of our circle.

### The Right-Angled Triangle

To find the length of any chord, we do not need to find the intersection points directly. Instead, we use the geometric property of chords.

If you draw a perpendicular line from the center of the circle to the chord, it bisects the chord. This creates a right-angled triangle where the hypotenuse is the radius $r$, one leg is the perpendicular distance $d$ from the center to the line, and the other leg is half the chord length, $\frac{L}{2}$.

By the Pythagorean theorem:

$$ \left( \frac{L}{2} \right)^2 + d^2 = r^2 $$

This simplifies to the following expression for the chord length:

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

### Calculating the Distance

We need the perpendicular distance $d$ from the origin $(0,0)$ to the line $x + y - n = 0$. Using the standard formula for the distance from a point to a line, we get:

$$ d = \frac{|1(0) + 1(0) - n|}{\sqrt{1^2 + 1^2}} = \frac{n}{\sqrt{2}} $$

This distance $d$ tells us exactly how far each line is from the center. As $n$ grows, $d$ grows, and the chord length $L$ must shrink.

### The Master Equation

Now, let us square the chord length formula: $L^2 = 4(r^2 - d^2)$. Substituting $r^2 = 16$ and $d^2 = \frac{n^2}{2}$, we obtain:

$$ L^2 = 4\left(16 - \frac{n^2}{2}\right) = 64 - 2n^2 $$

This is our master equation. It tells us the squared length of any chord for a given $n$.

However, we have a constraint: a chord only exists if the line intersects the circle, meaning $d < r$. Thus:

$$ \frac{n}{\sqrt{2}} < 4 \Rightarrow n < 4\sqrt{2} \approx 5.65 $$

Since $n$ is a natural number, $n$ can only be $1, 2, 3, 4,$ or $5$.

### The Final Summation

Now, we calculate $L^2$ for each valid $n$:

For $n=1$: $L_1^2 = 64 - 2(1)^2 = 62$

For $n=2$: $L_2^2 = 64 - 2(2)^2 = 56$

For $n=3$: $L_3^2 = 64 - 2(3)^2 = 46$

For $n=4$: $L_4^2 = 64 - 2(4)^2 = 32$

For $n=5$: $L_5^2 = 64 - 2(5)^2 = 14$

Adding these values together:

$$ 62 + 56 + 46 + 32 + 14 = 210 $$

The final result for the sum of the squares of the lengths of the chords is **210**.

Animated Solution for Mathematics - Circles: The sum of the squares of the lengths of the chords intercepted on the circle, $x^{2} + y^{2} = 16$ , by the lines, $x + y = n, n \in N$ , where $N$ is the set of all natural numbers, is :

Select Answer:

Visualized Solution

Visualize the Circle

The Family of Lines

Chord Length Geometry

Distance from Origin

Squaring the Chord Length

Constraint on $n$

Calculating $L^{2}$ for $n = 1, 2, 3$

Calculating $L^{2}$ for $n = 4, 5$

Summing it all up

The Sigma Insight: Intercepts Made by a Circle

Analyzing the Setup

The Right-Angled Triangle

Calculating the Distance

The Master Equation

The Final Summation

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