Playbook
Straight Line
46 q - 0.96/paper - 22% HARD. Two broad subtopics, both routine coordinate geometry. Section Formula, Concurrency and Foot of Perpendicular (27 q) is 19% HARD and shares its whole toolkit with Line and Plane, so it costs almost nothing on top of a cornerstone you already own.
- Questions in the bank
- 46
- q/paper (2024–25 shifts)
- 0.96
- Tagged HARD
- 22%
- Subtopics
- 2
Strand: Quick-Win
When you’ll see it
A line in the plane: a point dividing a segment, a distance from a point, three lines tested for concurrency, an angle between two lines, or a bisector.
How this chapter is tested
46 q at 0.96 per paper and 22% HARD. Two broad subtopics, both routine coordinate geometry, and the reason this sits in the Quick-Win strand rather than the long tail is not its size — it is that it shares its whole toolkit with Line and Plane, a cornerstone worth 4.96 questions a paper. Section formula, foot of perpendicular, distance from a point, angle between two objects: every one of these is the 2-D version of a move you already have to own. Once the cornerstone is yours, this chapter costs almost nothing on top.
Section Formula, Concurrency, Foot of Perpendicular, and Distance is 27 q at 19% HARD, the cheaper of the two blocks. Equation of Line — Rotation, Angle, and Bisector is 19 q at 26%. Neither is expensive, and there is no HARD concentration to route around.
One idea in here pays well beyond this chapter. Concurrency of three lines is a vanishing 3x3 determinant, and the survey found that same determinant working as a universal degeneracy test across five or six chapters — as collinearity of three points, as coplanarity of lines in three dimensions, and as scalar triple product zero in Vectors. Learn the test once and recognise its four costumes.
The same is true of the angle condition. The survey counted the angle-between-two-objects idea across 87 q in 7 chapters and perpendicularity across 83 q in 7. Here it is the slope-product condition; in Vectors it is a dot product; in Pair of Straight Lines it is a coefficient sum. One idea, four dialects.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Section formula, distance, and the foot of perpendicular
Internal and external division, the distance from a point to a line, and dropping a perpendicular onto a line. 27 q at 19% HARD — the cheapest block, and directly transferable to Line and Plane.
Concurrency of three lines
Three lines are concurrent when the determinant of their coefficients vanishes. The same test reads as collinearity of points and coplanarity in three dimensions. Part of the 27 q block.
Equation of a line in its standard forms
Slope-intercept, point-slope, two-point, intercept and normal forms, and moving between them to match whichever form the options use. Part of the 19 q, 26% HARD subtopic.
Angle between lines, bisectors, and rotation
The slope-product condition for perpendicularity, the tangent formula for a general angle, both angle bisectors, and the effect of rotating a line about a point. 19 q at 26% HARD.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Perpendicularity applied to a vertical line
The slope-product condition needs both slopes to exist. A line parallel to the y-axis has no slope, and the formula silently produces a wrong answer rather than failing.
The other angle bisector
Two lines have two bisectors, at right angles to each other, and both appear in the options. The question usually specifies the one containing the origin or the acute one — read which.
Distance formula without the modulus
The point-to-line distance is a non-negative quantity. The signed value, which the algebra produces first, is offered as a distractor and is the negative of the right answer.
Concurrency claimed for parallel lines
A vanishing determinant is necessary but does not by itself place three lines through one point — a set of parallel or coincident lines can satisfy it. Check that the lines actually meet.
Drill every straight line question
46 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.