Playbook
Pair of Straight Lines
45 q - 1.00/paper - 40% HARD. A closed, formula-driven chapter. Every question reduces to reading a, h and b out of a combined equation and applying one of a short list of conditions, which makes it more learnable than its 40% suggests.
- Questions in the bank
- 45
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 40%
- Subtopics
- 2
Strand: Long Tail
When you’ll see it
A homogeneous second-degree expression in x and y, or a general second-degree equation asked whether it represents two straight lines.
How this chapter is tested
45 q, 1.00/paper, 40% HARD — and the 40% overstates how hard it is to PREPARE. This is the most closed chapter on the paper: essentially every question reduces to reading a, h and b out of a combined equation and applying one item from a short list of conditions. The two subtopics behave alike (Combined Equation and Condition, 28 q at 39%; Angle, Distance, and Geometry of Pair, 17 q at 41%), so there is no cherry-picking, but equally no surprise.
The conditions are the cross-chapter angle family speaking this chapter's dialect. Perpendicularity is a + b = 0 here, where straight lines say m1 times m2 equals -1, vectors say the dot product is zero, and Line and Plane says it through the direction vectors and the plane normal. A survey of the bank puts perpendicularity at 83 q across 7 chapters and the parent idea, angle between two objects, at 87 q across 7 — so the formula learned here is being tested four more times under other names.
The other reusable piece is the degeneracy test. A general second-degree equation represents a pair of lines exactly when a particular 3x3 determinant vanishes — the same universal condition that shows up as concurrency of three lines, collinearity of three points, coplanarity, and a zero scalar triple product, measured across five to six chapters and roughly 19 to 30 questions.
Practical upshot: this is the tail chapter with the best ratio of preparation time to reliability. A checklist of six conditions, drilled once, holds up across all 45 questions.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Reading a, h and b
For ax^2 + 2hxy + by^2 the coefficient of xy is 2h, so h is HALF of what is printed. Every other formula in the chapter depends on getting this right.
Condition for real, distinct lines
h^2 greater than ab gives two distinct real lines, h^2 equal to ab gives coincident lines, and h^2 less than ab gives no real lines (only the origin).
Separating the pair
Factorise the homogeneous expression into two linear factors, or solve it as a quadratic in y/x to get the two slopes. The slopes satisfy sum = -2h/b and product = a/b.
Angle between the pair
tan theta is the modulus of 2 times the square root of (h^2 - ab), all over (a + b). Perpendicular exactly when a + b = 0; coincident exactly when h^2 = ab.
The general second-degree case
Before applying any homogeneous result to an equation carrying x, y or constant terms, check the 3x3 determinant condition. If it does not vanish, the equation is a conic, not a pair.
Distances and bisectors
Distance between the two lines of a parallel pair, and the combined equation of the angle bisectors of the pair. Short, formula-driven, and the whole of the smaller subtopic.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
2h read as h
The single commonest slip in the chapter. Using the printed xy coefficient as h doubles it, and every downstream answer — angle, condition, slopes — lands on a supplied wrong option.
Perpendicularity confused with h = 0
a + b = 0 is the perpendicular condition. h = 0 only means the pair is symmetric about the axes, which is a different statement entirely.
Coincident read as non-existent
h^2 = ab gives two coincident real lines, not zero lines. The option asserting no lines exist is the trap.
Homogeneous formula on a general equation
Applying the angle or perpendicularity condition to an equation with linear terms, without first verifying the determinant condition, answers a question that was never asked.
Drill every pair of straight lines question
45 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.