Page 2: Practice Problems and Multiple Circle Scenarios

This page focuses on applying the równanie okręgu wzory (circle equation formulas) to solve various problems involving circles in coordinate systems.

The first problem demonstrates how to find the coordinates of a circle's center given its equation:

Example: Given the circle equation $x+3$² + $y-4$² = 25, determine the coordinates of the center S.

The solution process is shown, leading to the center coordinates S$-3,4$.

The second problem introduces a more complex scenario involving two circles:

1. $x+1$² + $y-2$² = 9
2. x² + y² = 10

Students are asked to calculate the distance between the centers of these circles.

Highlight: This problem combines the concepts of circle equations and distance calculation in the coordinate plane.

The page provides a step-by-step approach to solving this problem, including identifying the centers of both circles: S₁$-1,2$ and S₂(0,0).

Page 3: Distance Calculation and Problem Solution

This page concludes the problem from the previous page, focusing on the distance calculation between two points in a coordinate system.

The okrąg w układzie współrzędnych wzory (formulas for circles in coordinate systems) are extended to include the distance formula:

Formula: |AB| = √((xв - xA)² + (yв - yA)²)

This formula is used to calculate the distance between the centers of the two circles from the previous problem:

Example: Distance between S₁$-1,2$ and S₂(0,0) is calculated as √$0-(-1)$² + $0-2$² = √5

The final answer is determined to be √5, which corresponds to option A in the multiple-choice question.

Highlight: This problem demonstrates the practical application of równanie okręgu wzory (circle equation formulas) and distance calculations in solving complex geometric problems in coordinate systems.

Page 1: Circle Equation Fundamentals

This page introduces the fundamental concepts of okrąg w układzie współrzędnych (circles in coordinate systems) and presents the równanie okręgu wzory (circle equation formulas).

The general form of the równanie okręgu w postaci kanonicznej (canonical form of circle equation) is presented as $x-a$² + $y-b$² = r², where (a,b) represents the circle's center and r is the radius.

Definition: The point S(a,b) represents the center of the circle in the coordinate system.

An example is provided to illustrate the application of the formula:

Example: For a circle with center S(2,4) and radius 3, the equation is $x-2$² + $y-4$² = 3².

The page includes a visual representation of this circle in the coordinate plane, helping students understand the relationship between the equation and its graphical representation.