Coordinate Geometry

Coordinate Geometry, also known as analytical geometry, bridges algebra and geometry by graphing points, lines, and curves on a Cartesian plane.

The Cartesian Plane: A 2D plane formed by two perpendicular number lines, the horizontal x-axis and the vertical y-axis. Their intersection is the origin (0,0).
Coordinates: A point is located using an ordered pair (x, y), where 'x' is the horizontal distance (abscissa) and 'y' is the vertical distance (ordinate) from the origin.

These formulas are the core tools for solving coordinate geometry problems.

Distance Formula: The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Section Formula: The coordinates of a point dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio m:n is \( (\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}) \).
Mid-Point Formula: A special case of the section formula where m:n = 1:1. The midpoint is \( (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) \).
Slope of a Line (m): The steepness of a line. \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Equations of a Line:
- Slope-Intercept Form: \( y = mx + c \) (c is the y-intercept)
- Point-Slope Form: \( y - y_1 = m(x - x_1) \)

Collinear Points: Three points are collinear (lie on the same straight line) if the slope between any two pairs of points is the same. Alternatively, the area of the triangle formed by them is zero.
Parallel & Perpendicular Lines: Two lines are parallel if their slopes are equal (\( m_1 = m_2 \)). They are perpendicular if the product of their slopes is -1 (\( m_1 \times m_2 = -1 \)).
Sketch it Out: Always draw a rough sketch of the points and lines on a coordinate plane. It helps in visualizing the problem and avoiding errors.

Memorize all the core formulas, especially the distance, section, and slope formulas.
Be extremely careful with positive and negative signs during calculations.
Many questions can be solved faster by visualizing the geometry rather than jumping straight into formulas.