Straight Lines
PromiseLines full marks: write a convenient form → apply distance/angle/bisector/family toolkit with sign care → verify with a known point.
01. Distance, section, and midpoint formulae are used constantly inside circle/conic stems too.
02. Angle bisector signs (±) are a classic trap — interpret which bisector the question wants.
03. Family L₁+λL₂=0 is the fastest way to force a line through an intersection.
How Advanced asks this
- Point–line distance / angle between lines
- Angle bisectors
- Family through intersection (L₁+λL₂)
- Concurrency / triangle centres
- Pair of straight lines conditions
Traps that leak marks
- Bisector sign errors
- Dropping absolute value in distance
- Wrong λ in family
- Acute/obtuse mix-up
- Using 1+m₁m₂=0 when lines are vertical/undefined slope without care
- Bisector ± chosen without a test point
- Distance formula missing absolute value or √(a²+b²)
Must cover (this topic)
- All standard line forms + angle bisectors
60–90s process
- 0–15s — Name ask: distance / angle / bisector / family / centre.
- 15–35s — Select the one toolkit formula; plug with absolute values where required.
- 35–75s — Compute carefully. For L₁+λL₂, fix λ from the extra condition.
- 75–90s — Verify with a known point or perpendicular check (1+m₁m₂=0).
Methods that change the score
Lines are pure toolkit with sign discipline. These methods make distance/bisector/family feel boringly repeatable — which is exactly how you bank marks.
Method 01 · Form match
Stops mixing slope formulae with ax+by+c coefficients — silent wrong answers.
Do this: Convert to the form your formula expects before substituting.
Method 02 · Test-point bisector
Resolves ± confusion that Advanced builds into bisector options.
Do this: After writing ± bisectors, test which one matches the required region.
Method 03 · L₁+λL₂ lock
Skips unnecessary intersection coordinates — pure speed.
Do this: Write family first; use extra condition (parallel to…, through…) to find λ.
Method 04 · Perpendicular shortcut
1+m₁m₂=0 saves the tanθ computation and division-by-zero panic.
Do this: If 1+m₁m₂=0, skip tanθ.
Quick formulas
- Distance between points: √[(x₂−x₁)²+(y₂−y₁)²] — Segment length
- Section (internal): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) — Division in ratio m:n
- Point–line distance: |ax₀+by₀+c|/√(a²+b²) — Distance from point to line
- Angle (slopes): tanθ=|(m₁−m₂)/(1+m₁m₂)| — Angle between two non-perpendicular lines
- Perpendicular: m₁m₂=−1 (finite slopes) — Right angle check
- Parallel distance: |c₁−c₂|/√(a²+b²) for ax+by+c₁=0 and ax+by+c₂=0 — Distance between parallel lines
- Bisectors: (L₁)/√(a₁²+b₁²)=±(L₂)/√(a₂²+b₂²) — Angle bisector equations
- Family: L₁+λL₂=0 — Lines through intersection of L₁=0, L₂=0
- Triangle area: (1/2)|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)| — Area / collinearity (area 0)
Common question families
Distance / angle toolkit
Looks like: Distance from point to line; angle between lines
Fast model: Write usable form → plug formula with absolute values / tanθ.
Family L₁+λL₂
Looks like: Line through intersection of two lines + extra condition
Fast model: Write L₁+λL₂=0 first; fix λ from condition — often skip finding intersection coordinates.
Angle bisectors
Looks like: Bisector equations; acute/obtuse bisector
Fast model: Write both ± forms; pick with a test point or angle condition.
Concurrency / pair of lines condition
Looks like: Three lines concurrent; combined equation represents pair of lines
Fast model: For concurrency use determinant of coeffs = 0; for pair of lines apply abc condition.
Sample preview · Distance / angle toolkit
Distance from (1,2) to the line x+y−1=0.
Answer: √2
Write the absolute value — don’t drop it.