Coordinate Geometry and Trigonometry
Lines, circles and trigonometry
Coordinate Geometry and Trigonometry tests straight-line geometry, the circle, trigonometric identities and equations, triangle solution, bearings, and radian measure. The exam favours questions where a diagram-free description must be converted into an equation or an angle.
Core rules
- Gradient m = (y2 - y1)/(x2 - x1). Parallel lines have equal gradients; perpendicular gradients satisfy m1 m2 = -1.
- Line forms: y = mx + c and y - y1 = m(x - x1). Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint = ((x1 + x2)/2, (y1 + y2)/2).
- Circle: (x - a)^2 + (y - b)^2 = r^2 has centre (a, b), radius r. A tangent is perpendicular to the radius at the point of contact.
- Identities: sin^2 A + cos^2 A = 1, tan A = sin A / cos A.
- Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a^2 = b^2 + c^2 - 2bc cos A. Area = (1/2)ab sin C.
- Bearings are measured clockwise from north and written with three digits. In radians, pi radians = 180 degrees.
Worked example
Find the equation of the tangent to x^2 + y^2 - 4x + 6y - 12 = 0 at P(5, 1).
Complete the square: x^2 - 4x = (x - 2)^2 - 4 and y^2 + 6y = (y + 3)^2 - 9. So (x - 2)^2 + (y + 3)^2 = 25. Centre C = (2, -3), radius 5. Gradient of CP: (1 - (-3))/(5 - 2) = 4/3. The tangent gradient is the negative reciprocal, -3/4. Using P(5, 1): y - 1 = (-3/4)(x - 5), so 3x + 4y = 19. Check: P satisfies it, and the tangent is perpendicular to the radius.
Traps
When a trig equation gives a negative sine or cosine, find all solutions in the stated interval, not just the calculator angle. Check whether the question is in degrees or radians before using pi. For bearings, start at north and turn clockwise; a back-bearing differs by 180 degrees. In a circle equation, the centre signs are opposite to those inside the brackets. Do not use the radius gradient as the tangent gradient.