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Kumar Rohan

Physics and Mathematics

Tangents & Normals — Complete Formula

⭐ – Most used in JEE
⚠️ – Common Mistake
💡 – Memory Hint

Slope of Tangent

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Slope	[m = \dfrac{dy}{dx}]	[m] = slope, [y] = dependent variable, [x] = independent variable	Instantaneous slope ⭐

💡 Memory Hint:
Derivative gives slope of tangent

Equation of Tangent

Point Form

Concept	Formula	Symbols Meaning	Key Notes
Tangent	[y – y_1 = m(x – x_1)]	[(x_1, y_1)] = point of contact, [m] = slope at that point	Most used ⭐

General Form

Concept	Formula	Symbols Meaning	Key Notes
Tangent	Substitute [y = f(x)] in point form	[f(x)] = function	Standard method ⭐

Normal to the Curve

Slope of Normal

Concept	Formula	Symbols Meaning	Key Notes
Slope	[m_n = -\dfrac{1}{m}]	[m_n] = slope of normal, [m] = slope of tangent	Negative reciprocal ⭐

Equation of Normal

Concept	Formula	Symbols Meaning	Key Notes
Normal	[y – y_1 = m_n(x – x_1)]	[(x_1, y_1)] = point	Direct application ⭐

💡 Memory Hint:
Normal ⟂ tangent

Parametric Form

Concept	Formula	Symbols Meaning	Key Notes
Slope	[\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}]	[t] = parameter	Important ⭐
Tangent	Use slope in point form	—	Standard approach

Implicit Function

Concept	Formula	Symbols Meaning	Key Notes
Slope	Differentiate implicitly	[y] depends on [x]	Solve for [\dfrac{dy}{dx}] ⭐

Angle Between Two Curves

Concept	Formula	Symbols Meaning	Key Notes
Angle	[\tan\theta = \left( \dfrac{m_1 – m_2}{1 + m_1 m_2}\right)]

Condition of Parallel & Perpendicular Tangents

Concept	Formula	Symbols Meaning	Key Notes
Parallel	[m_1 = m_2]	—	Same slope ⭐
Perpendicular	[m_1 m_2 = -1]	—	Negative reciprocal ⭐

Subtangent and Subnormal

Concept	Formula	Symbols Meaning	Key Notes
Subtangent	[\dfrac{y}{dy/dx}]	[y] = ordinate	Important ⭐
Subnormal	[y \cdot \dfrac{dy}{dx}]	—	Less frequent

💡 Memory Hint:
Subtangent = y / slope

Length of Tangent & Normal

Concept	Formula	Symbols Meaning	Key Notes
Tangent Length	[\left( y \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}\right)]
Normal Length	[\left( y \sqrt{1 + \dfrac{1}{(dy/dx)^2}}\right)]

Derivative at a Point (Alternate Form)

Concept	Formula	Symbols Meaning	Key Notes
Slope at [x = a]	[m = f'(a)]	—	Direct substitution ⭐

Special Cases

Concept	Formula	Symbols Meaning	Key Notes
Horizontal Tangent	[\dfrac{dy}{dx} = 0]	—	Flat line ⭐
Vertical Tangent	[\dfrac{dx}{dy} = 0]	—	Infinite slope ⚠️

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