Determining Whether a Line is a Chord

Introduction

Key Concepts

1. Definitions and Fundamental Concepts

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the centre of the circle. The distance from the centre to any point on the circle is known as the radius ($r$). A chord is a straight line segment whose endpoints both lie on the circle. Unlike the diameter, which is a special type of chord passing through the centre, a chord does not necessarily pass through the centre.

2. Conditions for a Line to be a Chord

For a line to qualify as a chord of a circle, it must satisfy specific conditions based on its position relative to the circle. The primary condition is that the line must intersect the circle at exactly two distinct points. If the line is tangent to the circle, touching it at only one point, it is not considered a chord but a tangent.

Mathematically, consider the standard equation of a circle: $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the centre and $r$ is the radius. A straight line can be represented in the form: $$ y = mx + c $$ To determine if this line is a chord, we substitute the linear equation into the circle's equation and solve for $x$. The discriminant of the resulting quadratic equation helps ascertain the nature of their intersection: - If the discriminant ($D$) is positive, there are two real and distinct points of intersection, indicating that the line is a chord. - If $D = 0$, the line is tangent to the circle. - If $D < 0$, the line does not intersect the circle, and hence, is neither a chord nor a tangent.

3. Calculating the Distance from the Centre to the Line

Another method to determine if a line is a chord involves calculating the perpendicular distance ($d$) from the centre of the circle to the line. The formula for this distance is: $$ d = \frac{|A h + B k + C|}{\sqrt{A^2 + B^2}} $$ where the line is given in the general form $Ax + By + C = 0$, and $(h, k)$ is the centre of the circle.

- If $d < r$, the line intersects the circle at two points, making it a chord. - If $d = r$, the line is tangent to the circle. - If $d > r$, the line does not intersect the circle.

4. Length of the Chord

Once confirmed that a line is a chord, calculating its length involves the following formula: $$ \text{Length of chord} = 2 \sqrt{r^2 - d^2} $$ where $d$ is the perpendicular distance from the centre to the chord.

This formula is derived from the Pythagorean theorem applied to the right triangle formed by the radius, the perpendicular from the centre to the chord, and half of the chord itself.

5. Example Problem

*Example:* Given a circle with centre at $(2, -3)$ and radius $5$. Determine whether the line $3x + 4y + 10 = 0$ is a chord of the circle.

First, calculate the perpendicular distance ($d$) from the centre $(2, -3)$ to the line $3x + 4y + 10 = 0$: $$ d = \frac{|3(2) + 4(-3) + 10|}{\sqrt{3^2 + 4^2}} = \frac{|6 - 12 + 10|}{5} = \frac{4}{5} = 0.8 $$ Since $d = 0.8 < 5 = r$, the line intersects the circle at two points, confirming it is a chord.

6. Graphical Interpretation

Graphically, a chord appears as a straight line cutting across the circle, connecting two points on its circumference. The position and slope of the chord relative to the circle's centre determine its length and whether it is a diameter.

Understanding the graphical representation aids in visualizing problems related to chords, tangents, and diameters, facilitating better problem-solving strategies in coordinate geometry.

7. Relationship with Other Geometric Elements

Chords are intrinsically linked to other geometric elements within a circle, such as diameters, radii, and tangents. They serve as foundational elements in various theorems and problem-solving techniques, including:

• Perpendicular Bisector Theorem: The perpendicular bisector of a chord passes through the centre of the circle.
• Intersecting Chords Theorem: When two chords intersect inside a circle, the products of their respective segments are equal.
• Power of a Point: Relates the lengths of tangents and secants from a common external point to the circle.

8. Practical Applications

Understanding chords has practical applications in various fields such as engineering, architecture, and even in creating artistic designs. For instance, determining the optimal placement of supports in a circular structure or designing curved segments in bridges often involves chord calculations.

Advanced Concepts

1. Theoretical Derivations and Proofs

Delving deeper into the concept of chords, it's essential to explore the theoretical underpinnings and proofs that solidify our understanding.

*Proof: The Perpendicular Bisector of a Chord Passes Through the Centre*

**Given:** A circle with centre $O$ and a chord $AB$.

**To Prove:** The line perpendicular to $AB$ at its midpoint $M$ passes through $O$.

**Proof:**

1. Let $M$ be the midpoint of chord $AB$.
2. Draw the perpendicular from $M$ to $AB$, and let it intersect the centre $O$ of the circle.
3. Triangles $OAM$ and $OBM$ are congruent (by RHS congruence: Right angle, Hypotenuse $OA = OB = r$, and Side $AM = BM$).
4. Therefore, $\angle OAM = \angle OBM$, implying that $O$, $M$, and the perpendicular line lie on the same straight path.
5. Hence, the perpendicular bisector of chord $AB$ passes through the centre $O$. $\square$

2. Complex Problem-Solving

*Problem:* In a circle with centre at $(1, 2)$ and radius $5$, find the equation of all chords that make an angle of $30^\circ$ with the horizontal axis.

**Solution:**

A chord making an angle of $30^\circ$ with the horizontal implies that its slope ($m$) is: $$ m = \tan(30^\circ) = \frac{\sqrt{3}}{3} $$ The general equation of the line is: $$ y - 2 = m(x - 1) $$ Substituting $m$: $$ y - 2 = \frac{\sqrt{3}}{3}(x - 1) \\ \Rightarrow \sqrt{3}x - 3y + \sqrt{3} + 6 = 0 $$ To ensure this line is a chord, compute the distance ($d$) from the centre $(1, 2)$ to the line: $$ d = \frac{|\sqrt{3}(1) - 3(2) + \sqrt{3} + 6|}{\sqrt{(\sqrt{3})^2 + (-3)^2}} = \frac{|\sqrt{3} - 6 + \sqrt{3} + 6|}{\sqrt{3 + 9}} = \frac{2\sqrt{3}}{\sqrt{12}} = \frac{2\sqrt{3}}{2\sqrt{3}} = 1 $$ Since $d = 1 < 5 = r$, the line is indeed a chord. Therefore, the equation of the chord is: $$ \sqrt{3}x - 3y + \sqrt{3} + 6 = 0 $$

3. Interdisciplinary Connections

The concept of chords extends beyond pure mathematics into fields like physics and engineering. For instance:

• Civil Engineering: Designing circular structures such as arches and bridges requires precise chord calculations to ensure structural integrity.
• Physics: Understanding the trajectories and forces within circular motion often involves chord-related computations.
• Computer Graphics: Rendering circular objects and their properties in digital environments utilizes geometric principles involving chords.

4. Analytical Geometry Approach

In analytical geometry, chords can be analyzed using vector methods and coordinate geometry to derive properties and solve complex problems.

*Example:* Find the midpoint of a chord defined by two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the circle.

The midpoint $M$ is given by: $$ M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$ This formula is pivotal in various applications, such as bisecting chords and establishing symmetrical properties within the circle.

5. Chords in Polygons

Chords play a significant role in the study of regular polygons inscribed in circles. The side of a regular polygon is a chord of the circumscribed circle. Understanding chord lengths helps in deriving formulas related to polygonal properties like area and perimeter.

*Example:* In a regular hexagon inscribed in a circle of radius $r$, each side is equal to the radius. This is because the central angle subtended by each side is $60^\circ$, leading to: $$ \text{Length of chord} = 2r \sin\left( \frac{60^\circ}{2} \right) = 2r \sin(30^\circ) = 2r \times \frac{1}{2} = r $$

6. Extension to Ellipses and Other Conic Sections

While chords in circles are well-defined, similar concepts extend to other conic sections like ellipses and hyperbolas. In an ellipse, a chord is defined similarly, connecting two points on the curve. However, due to the ellipse's varying radii, chord properties become more complex and involve additional parameters.

Comparison Table

Summary and Key Takeaways