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Playbook

Pair of Straight Lines

42 q - 1.00/paper - 43% HARD. A closed, formula-driven chapter in four notes pages. 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 41% suggests; the angle page (11 q, 64%) carries the cost.

Questions in the bank
42
q/paper (2024–25 shifts)
1.00
Tagged HARD
43%
Subtopics
4

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

42 q, 1.00/paper, 43% HARD — and the 41% 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. Four notes pages (/notes/mht-cet-maths/pair-of-straight-lines): the joint-equation page is 12 q at 17%, the slopes page 10 q at 50%, the angle page 12 q at 58%, the general-equation page 10 q at 40%.

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 ratios. The angle itself is tan θ = 2√(h² − ab)/|a + b|, and the pair through a point at a given angle is the Straight Line two-root problem multiplied out.

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 and coplanarity in three dimensions.

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 44 questions.

The sub-skills

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

  • Joint Equation of Two Lines — Product of Linear Factors and the Triangle They Form

    Multiply two lines to get the pair; factorise a pair to get the lines; with a third line, the triangle's vertices, centroid, median and circumcentre. 12 q at 17% HARD — the cheap page.

  • Slopes of a Homogeneous Pair — Sum, Product and Ratio Conditions

    For ax^2 + 2hxy + by^2 the coefficient of xy is 2h, so h is HALF of what is printed; the slopes satisfy sum = -2h/b and product = a/b, and a ratio m : n gives (m + n)² ab = 4mn h². 10 q at 50% HARD.

  • Angle Between the Pair — Perpendicular Pairs, Lines at a Given Angle and the Bisectors

    tan θ = 2√(h² − ab)/|a + b|; perpendicular exactly when a + b = 0, coincident when h² = ab; the pair at a given angle to a line by squaring the angle condition; the bisector pair (x² − y²)/(a − b) = xy/h. 11 q at 64% HARD.

  • General Second-Degree Equation — Condition for a Pair, Parallel Lines and Distances

    Before applying any homogeneous result to an equation carrying x, y or constant terms, check the 3x3 determinant condition; a parallel pair (h² = ab) factors as a perfect square and its gap is 2√((g² − ac)/(a(a + b))); the product of distances from a point. 9 q at 44% HARD.

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.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.

Pair of Straight Lines notes

Drill every pair of straight lines question

42 questions from the bank, scoped to 4 bundled subtopics.

Drill the 42 questions

Drill one subtopic at a time

The 4 subtopics this playbook covers, in catalog order.

  • Joint Equation of Two Lines — Product of Linear Factors and the Triangle They FormDrill
  • Slopes of a Homogeneous Pair — Sum, Product and Ratio ConditionsDrill
  • Angle Between the Pair — Perpendicular Pairs, Lines at a Given Angle and the BisectorsDrill
  • General Second-Degree Equation — Condition for a Pair, Parallel Lines and DistancesDrill

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