# Useful Formulae

8 COORDINATEGEOMETRY axes, lines, gradients and intersections A pair of axes imposed on the plane at right angles allow a point to be defined by a pair of real numbers opposite. The axes intersect at 0,0,the origin. Horizontal and vertical positions are often referred to as xand yrespectively. The equation of a line is given by y mx c where mis its gradient. This line cuts the yaxis at 0,c and the xaxis at ,0. A vertical line has a constant xvalue,taking the formx k. The line passing through x0 ,y0 with gradient n is given by the equation y nx y0 nx0. A line perpendicular to another of gradient n will have gradient . The equation of the line through x1 ,y1 and x2 ,y2 is y x x2 y2 when x1 x2. The angle between two lines,gradients mand n, satisfies tan In three dimensions,a zaxis is added and many equations take analogous forms. For instance,a sphere with radius rand centred at a,b,c is given by xa2yb2zc2 r2. The general equation for a plane in 3d is ax bycz d. n c m y2 y1 1 mn m n A circle,radius r, centre a,b is given by x a2 yb2 r2. x2 x1 9