Glossary
Center of a Circle
The fixed point (h, k) from which all points on the circle are equidistant.
Example:
If a circle's equation is (x+1)² + (y-5)² = 9, its Center of a Circle is at (-1, 5).
Completing the Square
An algebraic technique used to rewrite a quadratic expression into a perfect square trinomial, essential for converting a general circle equation into standard form.
Example:
To find the center and radius from an expanded circle equation like x² + 4x + y² - 6y = 12, you'll need to use Completing the Square.
Diameter
A line segment that passes through the center of a circle and has its endpoints on the circle, equal to twice the radius.
Example:
If a circle has a radius of 5, its Diameter is 10.
Distance Formula
A formula used to calculate the distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane: d = √((x₂ - x₁)² + (y₂ - y₁)²).
Example:
You can use the Distance Formula to find the radius of a circle if you know its center and a point on the circle.
Midpoint Formula
A formula used to find the coordinates of the midpoint of a line segment given its endpoints (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2).
Example:
If you're given the endpoints of a circle's diameter, the Midpoint Formula will help you find the circle's center.
Origin
The point (0, 0) on a coordinate plane, often simplifying a circle's equation to x² + y² = r² if it's the center.
Example:
A circle centered at the Origin with a radius of 6 would have the equation x² + y² = 36.
Point of Tangency
The single point where a tangent line intersects a circle.
Example:
If a line y = x touches the circle x² + y² = 2 at (1, 1), then (1, 1) is the Point of Tangency.
Point on a Circle
Any coordinate (x, y) that satisfies the circle's equation, meaning it lies exactly on the circumference.
Example:
To check if (3, 4) is a Point on a Circle with equation x² + y² = 25, substitute the coordinates and see if the equation holds true.
Radius
The distance from the center of a circle to any point on its circumference.
Example:
A circle with an equation of x² + y² = 49 has a Radius of 7.
Standard Form of a Circle Equation
The algebraic representation of a circle on a coordinate plane, given by (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Example:
To find the center and radius of a circle, always look for its Standard Form of a Circle Equation.
Tangent Line
A line that touches a circle at exactly one point, known as the point of tangency, and is perpendicular to the radius at that point.
Example:
The radius drawn to the point where a Tangent Line touches the circle will always form a 90-degree angle.