Coordinate Geometry

•  
• see also:
  • Algebra
  • Mensuration
  • Calculus
  • Circular functions
  • Finance equations
  • Logarithmic, exponential and hyperbolic functions
  • Probability theory

Straight Line:

• equation y = mx + c;
• gradient between two points (x₁,y₁) and (x₂,y₂) is m = (y₂ - y₁) / (x₂ - x₁) ie. "rise over run"
• Y-intercept = c;
• equation of perpendicular line intersecting at (a,b):
  • gradient m' = 1/m;
  • => y' = x'/m + (b - a/m);
• intersection of 2 lines:
  • Solve with simultaneous equations;

Quadratic Functions:

• equation: f(x) = ax² + bx + c, a <> 0;
• has a minimum value if a>0, a maximum value if a<0;
• when f(x)=0, x = (-b + SQR(D))/2a, and, x = (-b - SQR(D))/2a, where D = b² - 4ac;
• to solve quadratic functions where D is negative, requires use of complex numbers to give "imaginary" solutions

Circle:

• (x-h)² + (y-k)² = r², where r = radius, (h,k) centre;

Division of a Line Segment in a given ratio:

• if A,B are 2 points on a line whose coords are (x,y) & (x',y') respectively, then the coords. of the point which divides AB internally in the ration m:m' are:
  • ((mx'+m'x)/(m+m'), (my'+m'y)/(m+m'));

Ellipse:

• general equation: x² / a² + y² / b² = 1,
• foci: (+/-ae,0), DD': x = +/-a/e, 0<e<1,
• b² = a²(1-e²);

Hyperbola:

• general equation: x² / a² - y² / b² = 1,
• foci: (+/-ae,0), DD': x = +/-a/e, e>1,
• b² = a²(e²-1);

Parabola:

• general equation: y² = 4ax,
• foci: (a,0) DD': x = -a, e=1,