Coordinate Geometry

Basics of Coordinate Geometry:

Cartesian Coordinate System:

• Introduction: A system that uses numerical coordinates to represent points on a plane.
• Components: Consists of two perpendicular lines (axes): the x-axis (horizontal) and the y-axis (vertical) intersecting at the origin (0,0).
• Ordered Pairs: Representation of a point as (x, y), where x is the distance along the x-axis and y is the distance along the y-axis.

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: $\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

Various Shapes 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.