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Playbook

Straight Lines

A small toolkit — slope, distance, image — used two or three times in a row. The centres of a triangle are the most common setting.

Questions in the bank
112
q/paper in 2025–26
0.92
Numeric answer
18%
Notes pages
7

Tier: Long tail

When you’ll see it

Lines in the xy-plane: slopes, the distance from a point to a line, an image in a line, a centre of a triangle, or a locus.

How this chapter is tested

Straight Lines is coordinate geometry with a small toolkit: the slope, the angle formula tan θ = |(m₁ − m₂)/(1 + m₁m₂)|, the distance |ax₁ + by₁ + c|/√(a² + b²), and the image formula. Almost every question is two or three of these in a row.

The centres of a triangle are the most common setting — above all the orthocentre, found from two altitudes or given and worked back to a vertex. Forms of a line and the angle between lines, areas from coordinates, images and reflected rays, and distances between parallel lines follow. Bisectors, pairs of lines and locus are the smaller pages.

Few questions are hard. The cost is that many have two answers — two slopes from one modulus, two signs from one area, three positions for a parallelogram's fourth vertex — and the options test whether you kept both. Lines also carry Conic Sections and Complex Numbers, where a tangent or a locus ends as a line.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Slope, angle and forms of a line

    tan θ = |(m₁ − m₂)/(1 + m₁m₂)|; intercept form x/a + y/b = 1; normal form x cos α + y sin α = p.

  • Distance and parallel lines

    |ax₁ + by₁ + c|/√(a² + b²) from a point; |c₁ − c₂|/√(a² + b²) between parallel lines written with the same a and b.

  • Image of a point and reflected rays

    (x − x₁)/a = (y − y₁)/b = −2(ax₁ + by₁ + c)/(a² + b²); a reflected ray passes through the image of the source.

  • Angle bisectors and pairs of lines

    Bisectors: (a₁x + b₁y + c₁)/√(a₁² + b₁²) = ±(a₂x + b₂y + c₂)/√(a₂² + b₂²); for ax² + 2hxy + by² = 0, tan θ = |2√(h² − ab)/(a + b)|.

  • Centres of a triangle

    Orthocentre from two altitudes, circumcentre from two perpendicular bisectors, centroid as the mean of the vertices, incentre weighted by the opposite sides.

  • Area of triangles and quadrilaterals

    Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; in a parallelogram ABCD, A + C = B + D.

  • Family of lines and locus

    L₁ + λL₂ = 0 passes through the meeting point of L₁ and L₂; eliminate the parameter to get a locus.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Normal form without a unit normal

    3x + 4y = 10 must be divided by 5 first, giving p = 2. Reading p = 10 from the raw equation is wrong.

  • Foot for image

    With factor 1 the formula gives the foot of the perpendicular; with 2 it gives the image. The midpoint of a point and its image lies on the line.

  • One sign from a modulus

    An area of 4 means the bracket is 8 or −8, and a given angle gives two slopes. Dropping one loses half the answers.

  • Parallel lines not matched

    x + 2y + 1 = 0 and 2x + 4y + 7 = 0: double the first before using |c₁ − c₂|/√(a² + b²).

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Straight Lines notes

Drill every Straight Lines question

112 questions from the bank, across 7 subtopics.

Drill the 112 questions

Drill one subtopic at a time

The 7 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.