📚 Concepts

📚 Cartesian Coordinate System      

A plane with two perpendicular number lines (X-axis and Y-axis) that intersect at a point called the origin (0, 0).

📚 Quadrants      

The coordinate plane is divided into 4 quadrants:      

• Quadrant I (+𝑥, +𝑦)      

• Quadrant II (−𝑥, +𝑦)      

• Quadrant III (−𝑥, −𝑦)      

• Quadrant IV (+𝑥, −𝑦)

📚 Coordinates of a Point     

 Each point is represented as an ordered pair (𝑥, 𝑦), where:      

• 𝑥 → horizontal position      

• 𝑦 → vertical position

📚 Abscissa and Ordinate      

• Abscissa → 𝑥-coordinate of a point      

• Ordinate → 𝑦-coordinate of a point

📚 Distance Between Two Points      

Distance between points 𝐴(𝑥₁, 𝑦₁) and 𝐵(𝑥₂, 𝑦₂) is calculated using the distance formula.

📚 Section Formula      

Used to find the coordinates of a point that divides the line segment joining two points in a given ratio.

📚 Midpoint Formula      

A special case of the section formula when the ratio is 1:1. It gives the midpoint of a line segment.

✍️️ Formulas

✍️ Distance Formula      

To find the distance between two points 𝐴 (𝑥₁, 𝑦₁) and 𝐵 (𝑥₂, 𝑦₂):      

• 𝐷 = √ [(𝑥₂ − 𝑥₁)² + (𝑦₂ − 𝑦₁)²]

✍️ Section Formula (Internal Division)      

If point 𝑃 divides the line segment joining 𝐴 (𝑥₁, 𝑦₁) and 𝐵 (𝑥₂, 𝑦₂) in the ratio 𝑚:𝑛 internally, then:      

• 𝑃(𝑥, 𝑦) = { (𝑚𝑥₂ + 𝑛𝑥₁)/(𝑚 + 𝑛),  (𝑚𝑦₂ + 𝑛𝑦₁)/(𝑚 + 𝑛) }

✍️ Midpoint Formula      

The midpoint 𝑀 of the line joining 𝐴(𝑥₁, 𝑦₁) and 𝐵(𝑥₂, 𝑦₂):

• 𝑀 (𝑥, 𝑦) = {(𝑥₁ + 𝑥₂)/2,  (𝑦₁ + 𝑦₂)/2}

✍️ Special Case – Points on Axis      

• If a point lies on the X-axis → 𝑦 = 0 → Coordinates: (𝑥, 0)      

• If a point lies on the Y-axis → 𝑥 = 0 → Coordinates: (0, 𝑦)