Summary — What This Chapter Covers
- Gradient m = (y2 − y1)/(x2 − x1); line through (x1, y1): y − y1 = m(x − x1).
- Parallel lines have equal gradients; perpendicular gradients multiply to −1.
- Midpoint ((x1 + x2)/2, (y1 + y2)/2); distance √[(x2 − x1)2 + (y2 − y1)2].
- Circle with centre (a, b) and radius r: (x − a)2 + (y − b)2 = r2.
- Tangent ⟂ radius; the angle in a semicircle is 90°; the perpendicular bisector of a chord passes through the centre.
- Substituting a line into a curve gives a quadratic whose discriminant counts the intersections.
1. Straight Lines
Lines
Gradient, midpoint, distance
You can use any of y = mx + c, y − y1 = m(x − x1) or ax + by + c = 0. The point–gradient form is the quickest when you know one point and the gradient. The perpendicular bisector of AB passes through the midpoint of AB with gradient −1/mAB.
Perpendicular gradients
m1 × m2 = −1 ⇔ m2 = −1/m1
Worked Example
A is (1, 2) and B is (5, 10). Find the length of AB and the equation of the perpendicular bisector of AB.
Practice Question
Find the equation of the line through (2, −1) parallel to 3x − 4y = 7.
2. Circles
Circles
Centre, radius and tangents
The expanded form x2 + y2 + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g2 + f2 − c); in practice, complete the square in x and in y. Use circle geometry: the tangent at P is perpendicular to the radius to P, the angle in a semicircle is a right angle, and the perpendicular from the centre bisects a chord.
Circle
(x − a)2 + (y − b)2 = r2 · x2 + y2 + 2gx + 2fy + c = 0: centre (−g, −f), r = √(g2 + f2 − c)
Worked Example
The circle x2 + y2 − 6x + 4y − 12 = 0 passes through P(6, 2). Find its centre and radius, and the equation of the tangent at P.
Practice Question
A(−1, 4) and B(5, −4) are the ends of a diameter of a circle. Find its equation.
3. Intersections of Lines and Curves
Intersections
Substitute, then use the discriminant
To find where a line meets a curve, substitute the line into the curve. The resulting quadratic has two, one or no real roots, so the line cuts the curve at two points, touches it (is a tangent), or does not meet it.
Memorise