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: yy1=m(xx1) using a point (x1,y1) and slope m.
  • Two-Point Form: (yy1)=y2y1x2x1(xx1) using two points (x1,y1) and (x2,y2).
  • Parallel and Perpendicular Lines: Parallel lines have equal slopes, while perpendicular lines have negative reciprocal slopes.

Distance and Midpoint Formulae:

  • Distance Formula: d=(x2x1)2+(y2y1)2
  • Midpoint Formula: (x1+x22,y1+y22)

Various Shapes and Equations:

Circles:

  • Standard Form: (xh)2+(yk)2=r2, where (h,k) is the center and r is the radius.

Parabolas:

  • Standard Form: y=ax2+bx+c or x=ay2+by+c
  • Vertex Form: y=a(xh)2+k or x=a(yk)2+h

Ellipses:

  • Standard Form: (xh)2a2+(yk)2b2=1 for horizontal ellipses, (xh)2b2+(yk)2a2=1 for vertical ellipses.

Hyperbolas:

  • Standard Form: (xh)2a2(yk)2b2=1 for horizontal hyperbolas, (yk)2a2(xh)2b2=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.