Mathematics: Class 10 — Chapter 7: Coordinate Geometry

• To locate a point on a plane, a pair of perpendicular coordinate axes is required.
• The distance of a point from the y-axis is its x-coordinate or abscissa.
• The distance of a point from the x-axis is its y-coordinate or ordinate.
• Any point on the x-axis is represented as (x, 0).
• Any point on the y-axis is represented as (0, y).
• A linear equation of the form ax + by + c = 0 represents a straight line graphically.
• A quadratic equation of the form y = ax² + bx + c (where a ≠ 0) represents a parabola.

The distance between any two points P(x₁, y₁) and Q(x₂, y₂) is determined using the Pythagorean theorem:

Formula:

PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]

• This is known as the distance formula.
• The distance of a point P(x, y) from the origin O(0, 0) is given by: OP = √[x² + y²].
• Since distance is always non-negative, only the positive square root is considered.

Key Applications:

• Collinearity: Three points A, B, and C are collinear if the sum of the lengths of any two segments equals the third (e.g., AB + BC = AC).
• Triangle Classification: Verify if three points form an equilateral, isosceles, or right-angled triangle using side lengths.
• Quadrilateral Verification: Prove geometric shapes (like squares, rectangles, or rhombuses) by calculating side lengths and diagonal lengths.

The coordinates of a point P(x, y) which divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are given by:

Formula:

P(x, y) = ( (m₁x₂ + m₂x₁) / (m₁ + m₂), (m₁y₂ + m₂y₁) / (m₁ + m₂) )

• This is known as the section formula.
• If the division ratio is k : 1, the coordinates of P are: ( (kx₂ + x₁) / (k + 1), (ky₂ + y₁) / (k + 1) ).
• Mid-point Formula: If P is the mid-point of AB, it divides the segment in the ratio 1 : 1. The coordinates are: ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 ).

• The distance between P(x₁, y₁) and Q(x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²].
• The distance of P(x, y) from the origin is √[x² + y²].
• The section formula determines the coordinates of a point dividing a segment in a given ratio.
• The mid-point formula simplifies this ratio to 1:1.