Formulas for Lines and Circles

Distance Formula

The distance formula calculates the distance between two points in the coordinate plane.

• Formula:

• Application: Used to find the length of a segment between two points.

• Example: The distance between (1, 2) and (4, 6) is .

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points.

• Formula:

• Application: Useful for bisecting line segments or finding centers.

• Example: The midpoint between (2, 3) and (6, 7) is .

Slope of a Line

The slope measures the steepness and direction of a line.

• Formula:

• Equal slopes: Lines are parallel if their slopes are equal ().

• Perpendicular slopes: Lines are perpendicular if the product of their slopes is ().

• Example: The slope between (1, 2) and (3, 8) is .

Equations of Lines

Vertical and Horizontal Lines

• Vertical line: (all points have the same x-coordinate)

• Horizontal line: (all points have the same y-coordinate)

Slope-Intercept Form

The slope-intercept form expresses a line as , where is the slope and is the y-intercept.

• Example: has slope 2 and y-intercept 3.

Point-Slope Form

The point-slope form is used when a line passes through a specific point with slope .

• Formula:

• Example: A line through (2, 5) with slope 4:

General Form of a Line

The general form of a line is , where , , and are constants.

• Application: Useful for algebraic manipulation and identifying line properties.

Equations of Circles

Standard Form

The standard form of a circle with center and radius is:

• Example: A circle centered at (3, -2) with radius 5:

General Form

The general form of a circle is:

• Application: Used for algebraic manipulation and identifying circle properties.

• Restrictions: , , is arbitrary.

Review Exercises: Applications and Problem Types

Distance, Midpoint, and Slope Problems

• Find the distance, midpoint, and slope for given pairs of points.

• Example: For points (0, 4) and (2, 7): - Distance: - Midpoint: - Slope:

Intercepts and Graphing

• Find x- and y-intercepts of given equations.

• Sketch graphs of lines and circles using intercepts and other properties.

• Use graphing utilities to approximate intercepts and visualize equations.

Equation Construction

• Write equations of lines given slope and a point, or two points.

• Find equations of lines parallel or perpendicular to a given line through a specified point.

• Write equations of circles given center and radius, or other properties.

Circle Properties and Applications

• Find the center and radius from the general form of a circle.

• Show that points lie on a circle using the distance formula.

• Find equations for diameters and chords.

Key Comparison Table: Line and Circle Equations

Summary of Problem Types

• Distance, midpoint, and slope calculations

• Finding and interpreting intercepts

• Graphing lines and circles

• Writing equations from given information

• Analyzing relationships between lines (parallel, perpendicular)

• Circle properties and applications

Additional info: These notes expand on the brief formulas and exercise prompts in the original file, providing full academic context and examples for Precalculus students.