Core Concept
Slope & Equation
The slope of the tangent at point (x₁, y₁) on curve y = f(x) is m = f'(x₁) = dy/dx |_(x₁,y₁). The normal at the same point is perpendicular to the tangent, so its slope is –1/m (when m ≠ 0).
Equation of Tangent
y – y₁ = m(x – x₁)
where m = dy/dx at (x₁,y₁)
where m = dy/dx at (x₁,y₁)
Equation of Normal
y – y₁ = –(1/m)(x – x₁)
perpendicular to tangent
perpendicular to tangent
Special Cases
- 1Slope m = 0 : Tangent is horizontal (parallel to x-axis); Normal is vertical (parallel to y-axis).
- 2Slope m → ∞ : Tangent is vertical (parallel to y-axis); Normal is horizontal.
- 3Tangent at origin: If curve passes through (0,0), use (x₁,y₁) = (0,0) in the formula.
- 4Parallel tangents: Two tangents are parallel if they have the same slope.
- 5Perpendicular tangents: Slopes m₁ and m₂ satisfy m₁ · m₂ = –1.
Angle of Intersection
Angle between two curves at intersection
tan θ = |( m₁ – m₂ ) / ( 1 + m₁m₂ )|
Orthogonal curves: Two curves are orthogonal if their tangents at every point of intersection are perpendicular, i.e., m₁ · m₂ = –1.
Length of Tangent & Normal
If y₁ = f(x₁) and m = dy/dx at (x₁,y₁)
Length of Tangent = y₁ · √(1 + 1/m²) |
Length of Normal = y₁ · √(1 + m²)
Solved Examples
Example 01
Tangent to y = x² at (1, 1)
Q. Find equations of tangent and normal to y = x² at the point (1, 1).
y = x² → dy/dx = 2x
At (1,1): m = 2(1) = 2
Tangent: y – 1 = 2(x – 1) → y = 2x – 1
Normal: y – 1 = –(1/2)(x – 1) → 2y + x = 3
Tangent: y = 2x – 1 | Normal: x + 2y = 3
Example 02
Horizontal tangent on y = x³ – 3x
Q. Find points on y = x³ – 3x where the tangent is horizontal.
Horizontal tangent ⟹ dy/dx = 0
dy/dx = 3x² – 3 = 0 → x² = 1 → x = ±1
At x=1: y = 1–3 = –2 → Point (1, –2)
At x=–1: y = –1+3 = 2 → Point (–1, 2)
∴ Tangent is horizontal at (1, –2) and (–1, 2)
Example 03
Tangent to x² + y² = 25 at (3, 4)
Q. Find the equation of tangent to the circle x² + y² = 25 at (3, 4).
Differentiate implicitly: 2x + 2y·(dy/dx) = 0
dy/dx = –x/y → At (3,4): m = –3/4
Tangent: y – 4 = –(3/4)(x – 3)
4y – 16 = –3x + 9 → 3x + 4y = 25
∴ Equation of tangent: 3x + 4y = 25