Coordinate Geometry
Coordinate Geometry at Higher Level focuses on the line and the circle: equations, slopes, distances, midpoints, intersections, and tangents. Questions reward clear algebraic setup and accurate completion of the square.
Core rules
- Slope: (m=\frac{y_2-y_1}{x_2-x_1}). Equation of a line: (y-y_1=m(x-x_1)).
- Parallel lines have equal slopes. Perpendicular lines satisfy (m_1m_2=-1).
- Distance: (\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}). Midpoint: (\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)).
- Circle centre-radius form: ((x-h)^2+(y-k)^2=r^2).
- General form (x^2+y^2+2gx+2fy+c=0) has centre ((-g,-f)) and radius (\sqrt{g^2+f^2-c}).
- A tangent is perpendicular to the radius at the point of contact. For intersections, substitute the line into the circle; the discriminant shows whether there are two, one, or no real intersection points.
Worked example
Find the centre and radius of (x^2+y^2-4x+6y-12=0). Complete the square:((x-2)^2-4+(y+3)^2-9-12=0), so ((x-2)^2+(y+3)^2=25). The centre is ((2,-3)) and the radius is (5).
Traps
Dropping the negative signs when reading the centre from general form; forgetting to take the square root of the radius squared; using the radius slope instead of its negative reciprocal for a tangent; arithmetic slips in the distance formula; not checking whether an intersection is real.