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.
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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