Coordinate Geometry
What the domain tests
Coordinate geometry tests algebraic descriptions of straight lines and circles, intersection problems and loci. MSCE Additional Mathematics expects exact answers, often involving surds, and clear use of gradient and distance formulae rather than rough plotting.
Core rules
- Distance between (x1,y1) and (x2,y2) is √[(x2-x1)^2+(y2-y1)^2]. Midpoint is ((x1+x2)/2,(y1+y2)/2).
- Gradient m=(y2-y1)/(x2-x1). Parallel lines have equal gradients; perpendicular lines satisfy m1m2=-1.
- A line through (x1,y1) with gradient m is y-y1=m(x-x1). The general form is ax+by+c=0.
- A circle with centre (h,k) and radius r has equation (x-h)^2+(y-k)^2=r^2. Expanded, x^2+y^2+2gx+2fy+c=0 has centre (-g,-f) and radius √(g^2+f^2-c).
- A tangent at a point is perpendicular to the radius at that point.
Worked example
Find the equation of the circle with diameter AB, where A(1,2) and B(5,8). The centre is the midpoint: ((1+5)/2,(2+8)/2)=(3,5). The radius is half AB: 1/2√[(5-1)^2+(8-2)^2]=1/2√(16+36)=1/2√52=√13. Therefore (x-3)^2+(y-5)^2=13.
Traps
Using the wrong sign for the centre after completing the square; forgetting to halve the diameter; using the negative reciprocal for parallel lines; losing exact surd form by premature rounding.