MHT-CET Maths · Free notes
Pair of Straight Lines — MHT-CET Maths
Pair of Straight Lines is one question a paper and one of the more expensive chapters in MHT-CET Maths: two of every five of its past-year questions are HARD, and the difficulty is real algebra rather than a misread constraint. The whole chapter rests on one idea — the product of two linear equations is one quadratic equation, and the quadratic's coefficients remember the two slopes: their sum is −2h/b and their product a/b. From that come the slope-ratio conditions, the angle formula tan θ = 2√(h² − ab)/|a + b|, the bisector equation, and the determinant test that decides whether a general second-degree equation is a pair at all. The pages below run in that order, from writing a joint equation to reading a general one; the angle page holds the most HARD questions and the joint-equation page the cheapest ones. Every PYQ is tagged.
The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.
Subtopic notes
Joint Equation of Two Lines — Product of Linear Factors and the Triangle They Form
17 PYQsMultiply two linear equations to get the joint equation; factorise a joint equation to get the two lines back — and with a third line, the three lines bound a triangle whose vertices, centroid and circumcentre follow from the intersections.
Slopes of a Homogeneous Pair — Sum, Product and Ratio Conditions
16 PYQsFor ax² + 2hxy + by² = 0 the slopes satisfy m₁ + m₂ = −2h/b and m₁m₂ = a/b; any relation between the slopes (a ratio, a reciprocal, a common line) becomes an equation in a, h, b.
Angle Between the Pair — Perpendicular Pairs, Lines at a Given Angle and the Bisectors
21 PYQstan θ = 2√(h² − ab)/|a + b| is the angle between the two lines of ax² + 2hxy + by² = 0; a + b = 0 means perpendicular, h² = ab parallel, and the bisectors are (x² − y²)/(a − b) = xy/h.
General Second-Degree Equation — Condition for a Pair, Parallel Lines and Distances
19 PYQsax² + 2hxy + by² + 2gx + 2fy + c = 0 is a pair of lines iff abc + 2fgh − af² − bg² − ch² = 0 (with h² ≥ ab); when h² = ab the pair is parallel and the gap is 2√((g² − ac)/(a(a + b))).
Formula & revision sheet
12 formulas · 12 gotchas across all subtopics — the exam-eve cheat-sheet
Formula & revision sheet
12 formulas · 12 gotchas across all subtopics — the exam-eve cheat-sheet
Formulas (3)
Watch out for (3)
- 30° to the Y-axis read as slope tan 30°→ The Joint Equation Is the Product: (L₁)(L₂) = 0
- Reading xy − x + y − 1 as a curve→ Factorising a Pair: Split the Middle Term, or Complete the Square
- Finding A and B explicitly for the median→ The Triangle a Pair Makes With a Third Line: Vertices, Centroid, Median, Circumcentre
Formulas (3)
Watch out for (3)
- Dividing by a instead of b→ m₁ + m₂ = −2h/b and m₁m₂ = a/b
- Writing the identity with h² and ab swapped→ Slopes in a Ratio: (m + n)² ab = 4mn h², and the Reverse Direction
- Substituting the given line's own slope→ A Common Line Between Two Pairs, and a Line of the Pair Perpendicular to a Given Line
Formulas (3)
Watch out for (3)
- Using |a − b| in the denominator→ tan θ = 2√(h² − ab)/|a + b|: Perpendicular When a + b = 0, Parallel When h² = ab
- Sign of the xy term after substituting m = y/x→ The Pair Through a Point at a Given Angle to a Line: Square the Angle Condition
- Using the full xy coefficient as h→ The Angle Bisectors: (x² − y²)/(a − b) = xy/h
Formulas (3)
Watch out for (3)
- Counting k = 0 as a pair→ Condition for a Pair: abc + 2fgh − af² − bg² − ch² = 0
- Reporting the gap squared, or halving it→ Parallel Pair (h² = ab): Factor as a Perfect Square, or Use 2√((g² − ac)/(a(a + b)))
- Adding the distances→ Product of the Perpendicular Distances From a Point to the Two Lines
PYQ weightage by concept
12 concepts · 73 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
12 concepts · 73 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| The Joint Equation Is the Product: (L₁)(L₂) = 0 | 7 | 10% |
| The Triangle a Pair Makes With a Third Line: Vertices, Centroid, Median, Circumcentre | 4 | 5% |
| Factorising a Pair: Split the Middle Term, or Complete the Square | 1 | 1% |
| Concept | PYQs | Share |
|---|---|---|
| m₁ + m₂ = −2h/b and m₁m₂ = a/b | 4 | 5% |
| Slopes in a Ratio: (m + n)² ab = 4mn h², and the Reverse Direction | 4 | 5% |
| A Common Line Between Two Pairs, and a Line of the Pair Perpendicular to a Given Line | 2 | 3% |
| Concept | PYQs | Share |
|---|---|---|
| The Pair Through a Point at a Given Angle to a Line: Square the Angle Condition | 6 | 8% |
| tan θ = 2√(h² − ab)/|a + b|: Perpendicular When a + b = 0, Parallel When h² = ab | 3 | 4% |
| The Angle Bisectors: (x² − y²)/(a − b) = xy/h | 2 | 3% |
| Concept | PYQs | Share |
|---|---|---|
| Parallel Pair (h² = ab): Factor as a Perfect Square, or Use 2√((g² − ac)/(a(a + b))) | 5 | 7% |
| Condition for a Pair: abc + 2fgh − af² − bg² − ch² = 0 | 3 | 4% |
| Product of the Perpendicular Distances From a Point to the Two Lines | 1 | 1% |
Test yourself on Pair of Straight Lines
15 past MHT-CET questions from this chapter, timed at 27 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.