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maths:coordinate

a brief summary of coordinate geometry, parabolae, etc

Introduction

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;

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'));

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;

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,
maths/coordinate.txt · Last modified: 2021/07/24 11:39 by gary1