Distance & Section Formula
Distance: PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]
Section Formula (m:n): R = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) )
Internal & External division both covered.
Area & Centres of Triangle
Area = (1/2) | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |
Centroid (G)
G = ( (x₁+x₂+x₃)/3 , (y₁+y₂+y₃)/3 )
Incentre (I)
I = ( (ax₁+bx₂+cx₃)/(a+b+c) , (ay₁+by₂+cy₃)/(a+b+c) )
a, b, c are sides opposite to A, B, C
Equation of Straight Lines
| Form | Equation | Key Points |
|---|---|---|
| Slope-Intercept | y = mx + c | m = tan θ, c = y-intercept |
| Intercept | x/a + y/b = 1 | a = x-intercept, b = y-intercept |
| Normal | x cos α + y sin α = p | p = ⊥ distance from origin |
| Point-Slope | y − y₁ = m(x − x₁) | Through (x₁,y₁), slope m |
| Two-Point | (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁) | Through (x₁,y₁), (x₂,y₂) |
| Parametric | x = h + r cos θ y = k + r sin θ | From (h,k), angle θ |
Reflection, Foot & Distance from Line
Line: ax + by + c = 0
Image of (x₁,y₁)
(x−x₁)/a = (y−y₁)/b = −2(ax₁+by₁+c)/(a²+b²)
Foot of Perpendicular
(x−x₁)/a = (y−y₁)/b = −(ax₁+by₁+c)/(a²+b²)
Perpendicular Distance
|ax₁ + by₁ + c| / √(a² + b²)
Circle - Standard Forms
| Form | Equation | Centre | Radius |
|---|---|---|---|
| Standard | x² + y² = r² | (0,0) | r |
| General Centre | (x−h)² + (y−k)² = r² | (h,k) | r |
| General Equation | x² + y² + 2gx + 2fy + c = 0 | (−g, −f) | √(g² + f² − c) |
| Diameter Ends | (x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0 | Midpoint | Half distance |
Circle through 3 points → Determinant form = 0
Parametric: (h + r cos θ, k + r sin θ)
Position of Point w.r.t. Circle
S = x² + y² + 2gx + 2fy + c = 0
S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
S₁ > 0 → Outside S₁ = 0 → On S₁ < 0 → Inside
Tangent & Normal to Circle
For circle S = 0 → Tangent at (x₁,y₁): T = 0
T = xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
Tangent from external point → Pair of tangents: T² = SS₁
Key Highlights for JEE
Must know: All forms of line, Section formula, Image/Foot/Distance, General circle equation, Tangents, Diameter form, Parametric circle
Frequent questions: Locus, Pair of tangents, Chord of contact, Pole & polar, Radical axis, Orthogonality of circles
Weightage: 4–6 questions (12–18 marks) almost every year