maths:coordinate

# a brief summary of coordinate geometry, parabolae, etc

## 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):
• ⇒ 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'));

• 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, 