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.

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 $matriculation exam problems$.

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.

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.

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.

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.