Deriving Line Equations

This page delves deeper into the process of wyznacz równanie ogólne prostej przechodzącej przez punkty a i b (determining the general equation of a line passing through points A and B).

The page presents an exercise that guides students through the steps of finding the equation of a line given two points: A(-2,5) and B(4,0).

Example: Using the point-slope form of a line equation: (y - y₁)(x₂ - x₁) - (y₂ - y₁)(x - x₁) = 0

The solution is worked out step-by-step, demonstrating how to substitute the given points and simplify the equation.

Highlight: The final result, x + 2y - 4 = 0, is an example of the równanie ogólne prostej (general equation of a line).

This exercise reinforces the concept of transforming point information into a standard linear equation form, which is crucial for solving more complex geometric problems.

General Equation of a Line

This page focuses on the równanie ogólne prostej (general equation of a line) and its properties. The general form Ax + By + C = 0 is introduced and explained.

Definition: The general equation of a line is expressed as Ax + By + C = 0, where A, B, and C are constants and A and B are not both zero.

The page demonstrates how to interpret this equation by finding intercepts:

1. y-intercept: Set x = 0 and solve for y
2. x-intercept: Set y = 0 and solve for x

Example: For the equation 3x + 4y - 12 = 0, the y-intercept is (0, 3) and the x-intercept is (4, 0).

This representation is particularly useful when dealing with lines that are vertical or have undefined slope in the slope-intercept form.

Highlight: Understanding how to work with the general form of a line equation is crucial for solving more complex geometric problems and is often used in zadania maturalne (matriculation exam problems).

Distance from a Point to a Line

This page introduces the formula for calculating the odległość punktu od prostej (distance from a point to a line). This concept is crucial in many geometric applications.

Definition: The distance d from a point (x₀, y₀) to a line Ax + By + C = 0 is given by the formula: d = |Ax₀ + By₀ + C| / √(A² + B²)

The page provides two exercises to demonstrate the application of this formula:

1. Finding the distance from point P(1,3) to the line 8x - 15y + 3 = 0
2. Calculating the distance from point A(3,1) to the line 2x + 3y - 6 = 0

Example: For the second exercise, the solution is worked out step-by-step, resulting in a distance of 1/√13.

An additional exercise is presented, involving finding the distance from a point to a line defined by two points. This requires first deriving the general equation of the line before applying the distance formula.

Highlight: Mastering this formula and its applications is essential for solving various odległość punktu od prostej zadania (point-to-line distance problems) in geometry.

Parallel and Perpendicular Lines

This page focuses on identifying proste równoległe (parallel lines) and proste prostopadłe (perpendicular lines). It introduces the conditions for parallelism and perpendicularity in terms of line equations.

Definition: Two lines are parallel if and only if their slopes are equal, or in terms of general equations, if A₁B₂ - A₂B₁ = 0.

Definition: Two lines are perpendicular if and only if the product of their slopes is -1, or in terms of general equations, if A₁A₂ + B₁B₂ = 0.

The page provides exercises to practice identifying parallel and perpendicular lines:

1. Determining if 2x - 3y + 6 = 0 and 4x - 6y - 8 = 0 are parallel
2. Checking if 2x - 3y + 6 = 0 and 3x + 2y + 12 = 0 are parallel or perpendicular

Example: In the first exercise, the lines are shown to be parallel as -12 - 12 = 0 satisfies the parallelism condition.

Highlight: Understanding these conditions is crucial for solving problems involving proste prostopadłe i równoległe - zadania (exercises on perpendicular and parallel lines) in analytic geometry.

Practice Exercises

This final page provides a series of practice exercises that encompass all the concepts covered in the previous sections. These exercises are designed to reinforce understanding and apply the learned techniques.

The exercises include:

1. Finding the equation of a line given a point and slope
2. Determining the equation of a horizontal line passing through a given point
3. Converting between different forms of line equations
4. Calculating the distance from a point to a line
5. Finding the equation of a line passing through two points

Example: One exercise asks to find the equation of a line passing through points (4,-1) and (-2,1), demonstrating the application of the point-slope form.

Highlight: These exercises are representative of the types of problems students might encounter in zadania maturalne (matriculation exam problems) related to analytic geometry.

The page concludes with solutions to these exercises, providing students with immediate feedback and a chance to check their work.

Vocabulary: Terms like wzór na odległość punktu od punktu (formula for distance between two points) and warunek prostopadłości prostych (condition for perpendicularity of lines) are reinforced through these practical applications.

Introduction to Linear Equations

This page introduces the fundamental concepts of linear equations in analytic geometry. It focuses on the równanie kierunkowe prostej (slope-intercept form) of a line and its components.

Definition: The slope-intercept form of a line is given by y = ax + b, where 'a' represents the slope and 'b' is the y-intercept.

The page explains that 'a' in the equation is equal to tan α, where α is the angle the line makes with the x-axis. It then provides two exercises to demonstrate how to determine the equation of a line given specific information.

Example: For a line with a 45° angle and y-intercept of -2, the equation is y = x - 2, as tan 45° = 1.

The second exercise shows how to find the equation of a line passing through given points, emphasizing the process of calculating the slope and y-intercept.

Highlight: The page stresses the importance of understanding how to derive the equation of a line from various given conditions, a crucial skill in analytic geometry.