# TWO DIMENSIONAL COORDINATE GEOMETRY

Coordinate Geometry is the unification of algebra and geometry in which algebra is used in the study of geometrical relations and geometrical figures are represented by means of equations. The most popular coordinate system is the rectangular Cartesian system.

Coordinates of a point are the real variables associated in an order to describe its location in space. Here we consider the space to be two-dimensional. Through a point O, referred to as the origin, we take two mutually perpendicular lines XOX’ and YOY’ and call them x and y axes respectively. The position of a point is completely determined with reference to these axes by means of an ordered pair of real numbers (x, y) called the coordinates of P where |x| and |y| are the distances of the point P from the y-axis and the x-axis respectively. x is called the x-coordinate or the abscissa of P and y is called the y-coordinate or the ordinate of P.

### Basics of Two-Dimensional Coordinate Geometry:

#### Cartesian Coordinate System:

• Plane: Representation of a flat surface with two perpendicular number lines intersecting at a point called the origin (0,0).
• Axes: Two lines known as x-axis (horizontal) and y-axis (vertical) intersecting at right angles.
• Ordered Pairs: Points are represented as (x, y) where x is the distance along the x-axis and y is the distance along the y-axis.

#### Plotting Points:

• Coordinates: The x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position.
• Quadrants: The plane is divided into four quadrants, numbered counterclockwise from I to IV.

#### Equations of Lines:

• Slope-Intercept Form: $y=mx+c$, where $m$ is the slope and $c$ is the y-intercept.
• Point-Slope Form: $y-{y}_{1}=m\left(x-{x}_{1}\right)$ using a point $\left({x}_{1},{y}_{1}\right)$ and slope $m$.
• Two-Point Form: $\left(y-{y}_{1}\right)=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\left(x-{x}_{1}\right)$using two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$.
• Parallel and Perpendicular Lines: Parallel lines have equal slopes, while perpendicular lines have negative reciprocal slopes.

#### Distance and Midpoint Formulae:

• Distance Formula: $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$
• Midpoint Formula: $\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

#### Slope and Relationships:

• Slope: Describes the steepness of a line; .
• Vertical and Horizontal Lines: Vertical lines have undefined slope (x = constant), horizontal lines have zero slope (y = constant).

### Various Geometric Figures and Equations:

#### Circles:

• Standard Form: $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$, where $\left(h,k\right)$ is the center and $r$ is the radius.

#### Parabolas:

• Standard Form: $y=a{x}^{2}+bx+c$ or $x=a{y}^{2}+by+c$
• Vertex Form: $y=a\left(x-h{\right)}^{2}+k$ or $x=a\left(y-k{\right)}^{2}+h$

#### Ellipses:

• Standard Form: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}+\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$ for horizontal ellipses, $\frac{\left(x-h{\right)}^{2}}{{b}^{2}}+\frac{\left(y-k{\right)}^{2}}{{a}^{2}}=1$ for vertical ellipses.

#### Hyperbolas:

• Standard Form: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$ for horizontal hyperbolas, $\frac{\left(y-k{\right)}^{2}}{{a}^{2}}-\frac{\left(x-h{\right)}^{2}}{{b}^{2}}=1$ for vertical hyperbolas.

### Transformation of Graphs:

#### Translation:

• Shift: Moving a graph horizontally or vertically by adding or subtracting constants to the x or y coordinates.

#### Reflection:

• Mirror Image: Reflecting a graph over the x-axis, y-axis, or any other line.

#### Rotation:

• Changing Orientation: Rotating a graph by a certain angle around the origin or a point.

#### Scaling:

• Resizing: Enlarging or reducing the size of a graph proportionally.

### Real-Life Applications:

• Physics: Motion analysis, trajectory calculations.
• Engineering: Structural design, CAD (Computer-Aided Design).
• Economics: Supply and demand curves, optimization problems.