JEEClass 11-12
Coordinate Geometry Formulas for JEE
All Coordinate Geometry formulas for JEE — Straight Lines, Circle, Parabola, Ellipse, and Hyperbola. Conics typically contribute 2-3 questions per JEE paper.
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Straight Lines
Slope-intercepty = mx + c
Point-slopey − y₁ = m(x − x₁)
Two-point(y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)
Intercept formx/a + y/b = 1
Normal formxcosα + ysinα = p
Distance pt to lined = |ax₁ + by₁ + c| / √(a²+b²)
Angle between linestanθ = |m₁ − m₂| / |1 + m₁m₂|
Family of linesL₁ + λL₂ = 0
Circle
Standard formx² + y² = r²
General formx² + y² + 2gx + 2fy + c = 0, centre (−g,−f), r = √(g²+f²−c)
Tangent at (x₁,y₁)xx₁ + yy₁ = r²
Tangent (slope m)y = mx ± r√(1+m²)
Length of tangentL = √(x₁² + y₁² + 2gx₁ + 2fy₁ + c)
Power of a pointS₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
Parabola (y² = 4ax)
Parametric(at², 2at)
Focus(a, 0), Directrix: x = −a
Tangent at tty = x + at²
Tangent (slope m)y = mx + a/m
Normal at ty + tx = 2at + at³
Focal chordt₁t₂ = −1
Latus rectumLength = 4a
Ellipse (x²/a² + y²/b² = 1)
Parametric(acosθ, bsinθ)
Eccentricitye = √(1 − b²/a²), a > b
Foci(±ae, 0)
Tangent (slope m)y = mx ± √(a²m² + b²)
Tangent at θxcosθ/a + ysinθ/b = 1
Director circlex² + y² = a² + b²
Auxiliary circlex² + y² = a²
Hyperbola (x²/a² − y²/b² = 1)
Parametric(asecθ, btanθ)
Eccentricitye = √(1 + b²/a²)
Asymptotesy = ±(b/a)x
Tangent (slope m)y = mx ± √(a²m² − b²)
Rectangular hyperbolaxy = c², parametric: (ct, c/t)
Director circlex² + y² = a² − b² (if a > b)
JEE Tips
- Tip 1:For tangent problems, 'condition of tangency' (set discriminant = 0) is often the fastest approach
- Tip 2:Parametric form is almost always easier than Cartesian for conics
- Tip 3:Memorize the focal chord property t₁t₂ = −1 for parabola — it appears very frequently