Equations of Circles and Their Properties

Introduction

Key Concepts

Definition of a Circle

A circle is a set of all points in a plane that are equidistant from a fixed point known as the center. The constant distance from the center to any point on the circle is called the radius, denoted by \( r \).

Standard Equation of a Circle

The standard form of the equation of a circle with center at \( (h, k) \) and radius \( r \) is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ This equation represents all the points \( (x, y) \) that are at a distance \( r \) from the center \( (h, k) \).

Circumference and Area

Two fundamental properties of a circle are its circumference and area. The circumference \( C \) is the distance around the circle and is calculated by: $$ C = 2\pi r $$ The area \( A \) enclosed by the circle is given by: $$ A = \pi r^2 $$

General Equation of a Circle

A circle can also be represented by the general quadratic equation: $$ Ax^2 + Ay^2 + Bx + Cy + D = 0 $$ where \( A \neq 0 \). To convert this to the standard form, complete the squares for both \( x \) and \( y \) terms.

Completing the Square

To find the center and radius from the general equation, follow these steps:

1. Group the \( x \) and \( y \) terms:
   \( Ax^2 + Bx + Ay^2 + Cy = -D \)
2. Divide all terms by \( A \) if \( A \neq 1 \):
   \( x^2 + \frac{B}{A}x + y^2 + \frac{C}{A}y = -\frac{D}{A} \)
3. Complete the square for \( x \) and \( y \):
   \( \left(x + \frac{B}{2A}\right)^2 + \left(y + \frac{C}{2A}\right)^2 = \frac{B^2 + C^2}{4A^2} - \frac{D}{A} \)
4. Identify the center \( \left(-\frac{B}{2A}, -\frac{C}{2A}\right) \) and radius \( r = \sqrt{\frac{B^2 + C^2 - 4AD}{4A^2}} \)

Merged and Concentric Circles

Two circles are said to be concentric if they share the same center but have different radii. The equations for two concentric circles with center \( (h, k) \) and radii \( r_1 \) and \( r_2 \) are: $$ (x - h)^2 + (y - k)^2 = r_1^2 $$ and $$ (x - h)^2 + (y - k)^2 = r_2^2 $$ where \( r_1 \neq r_2 \).

Intersecting Circles

When two circles intersect, they do so at one or two points. The number of intersection points depends on the distance between their centers compared to the sum and difference of their radii:

• If the distance between centers \( d = 0 \) and radii are equal, the circles coincide.
• If \( d < |r_1 - r_2| \), one circle lies entirely within the other.
• If \( |r_1 - r_2| < d < r_1 + r_2 \), the circles intersect at two points.
• If \( d = r_1 + r_2 \), the circles touch externally at one point.

Tangent Circles

Two circles are tangent if they touch at exactly one point. Tangency can be either internal or external:

• External Tangency: The distance between centers \( d = r_1 + r_2 \).
• Internal Tangency: The distance between centers \( d = |r_1 - r_2| \).

Parametric Equations of a Circle

A circle can also be represented parametrically using an angle \( \theta \): $$ x = h + r \cos(\theta) $$ $$ y = k + r \sin(\theta) $$ where \( 0 \leq \theta < 2\pi \). This representation is particularly useful in calculus and when dealing with motion along a circular path.

Applications of Circle Equations

Understanding circle equations is crucial in various applications such as:

• Engineering: Designing circular structures and components.
• Physics: Analyzing circular motion and oscillations.
• Computer Graphics: Rendering circular shapes and animations.
• Navigation: Calculating distances and plotting courses.

Transformations of Circles

Circles can undergo various transformations such as translations, rotations, and scaling. However, because a circle is defined by all points equidistant from its center, rotations do not alter its appearance. Translations shift the circle's position without changing its size, while scaling changes its radius proportionally.

Equation of a Circle in Polar Coordinates

In polar coordinates, where a point is defined by its distance from the origin \( r \) and angle \( \theta \), the equation of a circle with center at \( (r_0, \theta_0) \) and radius \( a \) can be expressed as: $$ r^2 - 2ar \cos(\theta - \theta_0) + a^2 - r_0^2 = 0 $$ This form is particularly useful in fields like physics and engineering where polar coordinates simplify problem-solving.

Intersection with Axes

To find where a circle intersects the coordinate axes, set \( y = 0 \) to find \( x \)-intercepts and \( x = 0 \) to find \( y \)-intercepts. Solving the resulting equations will give the points of intersection, if any.

Comparison Table

Summary and Key Takeaways