Tangents & Normals 

Geometrical Interpretation of Derivative

• For a curve y = f(x), the derivative dy/dx represents the slope of the tangent.
• If the tangent makes an angle ψ with the positive x-axis:
• dy/dx = tan ψ

At any point (x, y) on y = f(x)

• Inclination of tangent: ψ = tan⁻¹(dy/dx)
• Slope of tangent: dy/dx
• Slope of normal: −dx/dy = −1/(dy/dx)

Equation of Tangent

Point Form

• Tangent at point (x₁, y₁):
• y − y₁ = (dy/dx)_(x₁,y₁) (x − x₁)

Special Cases

• Tangent parallel to x-axis ⇒ dy/dx = 0
• Tangent perpendicular to x-axis ⇒ dx/dy = 0
• Tangent equally inclined to both axes ⇒ dy/dx = ±1
• Value of ψ lies in (−π, π]

Parametric Form

• If x = f(t), y = g(t):
• dy/dx = g′(t)/f′(t)
• Equation of tangent at parameter t:
• y − g(t) = [g′(t)/f′(t)] (x − f(t))

Length of Tangent

• PT = y cosec ψ
• PT = y √(1 + cot²ψ)
• In terms of derivatives:
• PT = y √(1 + (dx/dy)²)

Length of Sub-Tangent

• TM = y cot ψ
• TM = y / (dy/dx)

Equation of Normal

Point Form

• Normal at point (x₁, y₁):
• y − y₁ = −(dx/dy)_(x₁,y₁) (x − x₁)
• Slope of normal: −1/(dy/dx)

Special Cases

• Normal parallel to y-axis ⇒ dy/dx = 0
• Normal parallel to x-axis ⇒ dx/dy = 0

Length of Normal

• PN = y sec ψ
• PN = y √(1 + tan²ψ)
• In terms of derivatives:
• PN = y √(1 + (dy/dx)²)

Length of Sub-Normal

• MN = y tan ψ
• MN = y (dy/dx)

Angle of Intersection of Two Curves

• If slopes at point of intersection are m₁ and m₂:
• tan ψ = |(m₁ − m₂)/(1 + m₁m₂)|
• Perpendicular intersection: m₁m₂ = −1
• Parallel curves: m₁ = m₂

Rolle’s Theorem

• If f(x) is defined on [a, b] such that:
• f is continuous on [a, b]
• f is differentiable on (a, b)
• f(a) = f(b)
• Then ∃ c ∈ (a, b) such that f′(c) = 0

Lagrange’s Mean Value Theorem (LMVT)

• If f(x) is continuous on [a, b] and differentiable on (a, b):
• Then ∃ c ∈ (a, b) such that:
• f′(c) = [f(b) − f(a)] / (b − a)

JEE Main Focus Tips

• Always compute dy/dx first
• Watch out for horizontal vs vertical tangents
• Length questions often mix geometry + derivatives
• Angle of intersection is a high-frequency application
• Rolle’s & LMVT conditions must be stated clearly in proofs