Coordinate Geometry
Core ideas
You need the equation of a straight line in several forms, distance and midpoint, perpendicular and parallel conditions, circle equations (including completing the square to find centre and radius), and the geometry of tangents and normals. Parametric curves and their Cartesian conversion are also examinable.
Traps
- The gradient of the normal is the negative reciprocal of the tangent, not just the negative.
- A line is tangent to a circle when the perpendicular distance from the centre to the line equals the radius; do not assume intersection alone proves tangency.
- When converting parametric to Cartesian, eliminate the parameter carefully and note any restriction on the domain.
- For a chord or tangent length, use right-triangle relationships with the radius, not raw coordinates.
Rule of thumb
For a circle x^2 + y^2 + 2gx + 2fy + c = 0, complete the square: centre (-g, -f), radius sqrt(g^2 + f^2 - c). Check the radius is real before proceeding.
Worked example
Find the tangent to the circle x^2 + y^2 - 4x + 6y - 12 = 0 at the point (5, 1). Completing the square: (x-2)^2 + (y+3)^2 = 25, centre (2, -3), radius 5. The radius gradient from (2,-3) to (5,1) is (1+3)/(5-2) = 4/3. The tangent gradient is -3/4. So y - 1 = -3/4 (x - 5), i.e. 3x + 4y = 19. Verify: distance from centre to line = |6 - 12 - 19|/5 = 25/5 = 5, equal to the radius.