Circle - Basic Concepts and Properties

1️⃣ Understand the definition of a circle and its basic components

2️⃣ Master the circle equation in both standard and general forms

3️⃣ Recognize special cases of circles

4️⃣ Solve practical problems involving circles and intersections

5️⃣ Apply circle properties to real-life problems

1️⃣ Definition of a Circle and Its Components

A circle is the locus of all points that are equidistant from a fixed central point. This fixed distance is called the radius, and the fixed central point is called the center of the circle.

Distance from any point on the circle to the center = radius (constant)

A circle consists of several basic elements: the center, radius, diameter, chord, and arc. Each element has its distinctive properties that determine the shape and size of the circle.

Important Note: The diameter equals twice the radius and is the longest chord in the circle

2️⃣ Circle Equation

The circle equation is the mathematical expression that describes all points located on the circle's circumference. There are two main forms for the circle equation.

Standard Form of Circle Equation

(x - h)² + (y - k)² = r²

where (h, k) is the center and r is the radius

3️⃣ Drawing a Circle on the Coordinate Plane

A circle can be drawn on the coordinate plane by knowing its center and radius. The center determines the circle's position, and the radius determines its size.

Drawing a circle with center (2, 1) and radius 3

Circle

Center (2,1)

Radius = 3

Steps to Find Circle Equation

Identify the center (h, k)

Determine the radius r

or calculate it from a known point

Apply the standard form

(x - h)² + (y - k)² = r²

4️⃣ Practical Examples with Circles

We will learn how to solve various problems involving circles, from finding equations to analyzing intersections.

Practical Examples

Example 1: Finding the Circle Equation

Find the equation of a circle with center (-2, 3) and radius 4

Solution:

Center: h = -2, k = 3
Radius: r = 4
Apply the formula: (x - h)² + (y - k)² = r²
(x - (-2))² + (y - 3)² = 4²
(x + 2)² + (y - 3)² = 16

Example 2: Finding Center and Radius

From the equation (x - 1)² + (y + 4)² = 25, find the center and radius

Solution:

Compare with standard form: (x - h)² + (y - k)² = r²
h = 1, k = -4 (note the sign!)
r² = 25 → r = 5
Center: (1, -4) and radius: 5

Example 3: Converting from General Form

Convert the equation x² + y² - 6x + 8y + 16 = 0 to standard form

Solution by completing the square:

x² - 6x + y² + 8y = -16
(x² - 6x + 9) + (y² + 8y + 16) = -16 + 9 + 16
(x - 3)² + (y + 4)² = 9
Center: (3, -4) and radius: 3

5️⃣ Circle Properties and Applications

Circles have important geometric properties that apply in many practical fields, from architectural engineering to space sciences.

Important Circle Properties:

Circumference = 2πr
Area = πr²
Any diameter divides the circle into two congruent halves
Inscribed angle = half the central angle
Tangent is perpendicular to radius at point of tangency

Summary of Key Points

1️⃣ A circle is the set of points equidistant from the center

2️⃣ Standard form: (x - h)² + (y - k)² = r²

3️⃣ Circles can be drawn knowing the center and radius

4️⃣ Completing the square converts general form to standard form

5️⃣ Circles have wide applications in practical life

Circle

Understanding Circle Standard Equation with Simulation