Equations of the Tangents and the Normals

1. Concept Overview

For a curve represented by [y=f(x)], the tangent at a point is a straight line that just touches the curve locally without cutting it (within a small neighborhood). The normal is the line perpendicular to the tangent at that same point.

Tangent → Shows instantaneous direction of the curve
Normal → Shows instantaneous perpendicular direction

This topic connects geometric slope (from coordinate geometry) with derivatives (from calculus).

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2. Derivation: Slope → Equation of a Tangent

At a point [P(x_{1},y_{1})] on the curve:

• The derivative gives slope of tangent:
  [m_{t} = \dfrac{dy}{dx}\bigg|_{x=x_{1}}]
• Using point-slope form, equation of tangent:
  [\boxed{y – y_{1} = m_{t}(x – x_{1})}]

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3. Equation of the Normal

Normal line slope = negative reciprocal of tangent slope

[m_{n} = -\dfrac{1}{m_{t}}]

Thus, Equation of the Normal:

[\boxed{y – y_{1} = -\dfrac{1}{m_{t}}(x – x_{1})}]

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4. When Curve is in Parametric Form

If:

[x = f(t), \quad y = g(t)]

At parameter [t=t_{0}], point is:

[x_{0}=f(t_{0}),;y_{0}=g(t_{0})]

Then slope:

[m_{t} = \dfrac{dy/dt}{dx/dt}\bigg|{t=t{0}}]

Tangent:

[\boxed{y – y_{0} = m_{t}(x – x_{0})}]

Normal:

[\boxed{y – y_{0} = -\dfrac{1}{m_{t}}(x – x_{0})}]

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5. Important Formulas to Remember (Table)

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6. Examples with Step-by-Step Solutions

Example 1

Find the tangent and normal to [y=x^{2}] at point [(2,4)].

Solution

1. [f(x)=x^{2} → f'(x)=2x]
2. At [x=2], slope of tangent: [m_{t}=4]
3. Tangent:
   [y – 4 = 4(x – 2)]
4. Slope of normal:
   [m_{n}=-\dfrac{1}{4}]
5. Normal:
   [y – 4 = -\dfrac{1}{4}(x – 2)]

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Example 2

Find the tangent to [y=\sin x] at [(\dfrac{\pi}{6},\dfrac{1}{2})].

1. [dy/dx = \cos x]
2. At [x=\dfrac{\pi}{6}],
   slope [m_{t}][=\cos(\dfrac{\pi}{6})][=\dfrac{\sqrt{3}}{2}]
3. Equation of tangent:
   [y – \dfrac{1}{2} ][= \dfrac{\sqrt{3}}{2}(x – \dfrac{\pi}{6})]

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Example 3

For parametric curve: [x=t^{2}, y=t^{3}], find tangent at [t=1].

1. [dx/dt = 2t], [dy/dt = 3t^{2}]
2. At [t=1], slope [m_{t}=\dfrac{3}{2}]
3. Point [(1,1)]
4. Tangent:
   [y – 1 = \dfrac{3}{2}(x – 1)]
5. Normal:
   [y – 1 = -\dfrac{2}{3}(x – 1)]

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Example 4

Find tangent to [x^{2}+y^{2}=25] at point [(3,4)].

(Implicit curve)

1. Differentiate:
   [2x + 2y\dfrac{dy}{dx}=0]
2. [\dfrac{dy}{dx} = -\dfrac{x}{y}]
3. At [(3,4)]: [m_{t} = -\dfrac{3}{4}]
4. Tangent:
   [y – 4 = -\dfrac{3}{4}(x – 3)]

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Example 5

Curve [y = \ln x], find tangent and normal at [x=e].

1. [dy/dx = \dfrac{1}{x}]
2. At [x=e], [m_{t}=\dfrac{1}{e}]
3. Point is [(e,1)]
4. Tangent:
   [y – 1 = \dfrac{1}{e}(x – e)]
5. Normal:
   [y – 1 = -e(x – e)]

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7. Key Insights

• Tangent = Best linear approximation to the curve near that point
• Normal = Direction of zero rate of change along tangent direction
• Together they form essential tools for further topics:
  • Curvature
  • Length of tangent/normal
  • Osculating circle

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8. Conceptual Questions With Solutions

1. Why does the derivative represent the slope of the tangent?

Because [\dfrac{dy}{dx}] gives the **instantaneous rate of change** of y with respect to x — the direction in which the curve moves at that exact point. A tangent line depicts the **best linear approximation** to that direction.

2. What happens if the derivative is zero at a point?

If [f'(a)=0], then the **tangent is horizontal** at [x=a]. The normal becomes a **vertical** line.

3. Can a tangent line intersect the curve at more than one point?

Yes. Example: A circle’s tangent touches at one point locally, but may intersect again elsewhere. Tangent definition is **local**.

4. Why is the slope of the normal the negative reciprocal of the tangent slope?

Because tangent and normal are **perpendicular**: [m_{t}m_{n} = -1] → slopes satisfy perpendicular condition.

5. Why must the point lie on the curve while finding tangent?

Because the tangent is defined **at that point** — it depends on the curve’s precise location and direction there.

6. What if the slope becomes infinite?

If [\dfrac{dy}{dx} \to \infty], then: Tangent → **vertical line** [x = constant] Normal → **horizontal line**

7. Can every point on a curve have a tangent?

No. If the curve has a **sharp corner or cusp** (e.g., |x| at x=0), derivative is undefined → no unique tangent.

8. Why do we sometimes use implicit differentiation?

Because curves like circles or ellipses are not explicitly given as [y=f(x)] — instead they are in relation form [F(x,y)=0].

9. Does the tangent depend only on slope?

No. It must also **pass through the specific point** — same slope at different points gives different tangents.

10. Why are tangents important in Physics?

They represent **instantaneous velocity direction** in motion along curved paths. Derivative = rate, tangent = direction.

11. What if a curve has multiple tangents at a point?

Then the curve has a **cusp**, **node**, or **intersection** — the direction is not uniquely defined.

12. Do normals always exist if tangent exists?

Yes, unless the tangent is **vertical** → slope infinite → normal slope = 0 (horizontal). Still valid.

13. Why does the tangent give best approximation near the point?

Using Taylor expansion: [ f(x) ≈ f(a) + f'(a)(x-a) ] → This is the tangent line — first-order approximation.

14. How does angle of inclination relate to derivative?

If θ is angle of tangent with x-axis: \[ [m_{t} = \tan \theta] \] Derivative directly gives geometric angle.

15. Why does tangent to a circle appear intuitive compared to other curves?

Because every radius is **perpendicular** to tangent at the point on circle → geometric symmetry simplifies derivative role.

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9. FAQ / Common Misconceptions

1. “Tangent means touching curve only once.”

False. A tangent is defined **locally**. Intersection points elsewhere do not change tangency.

2. “If f'(a) ≠ 0, tangent doesn’t exist.”

False. Tangent exists even if slope is non-zero. Zero slope only means **horizontal tangent**.

3. “Vertical tangent means no tangent exists.”

Incorrect. Derivative **infinite** slope → tangent is vertical: [x=a].

4. “If function continuous, tangent always exists.”

No — corners: continuous but **not differentiable** → no tangent.

5. “Normal only exists if tangent slope finite.”

Wrong. Normal **always exists** unless slope undefined due to non-differentiability.

6. “Slope is measured only algebraically, not geometrically.”

Incorrect. Slope = **tan θ**, a geometric angle measure.

7. “Equation of tangent doesn’t require the point.”

False — **point + slope** are essential.

8. “Implicit differentiation gives wrong slope.”

No — it allows correct slope for implicit curves like circles.

9. “Normal is always horizontal.”

Only if tangent is vertical.

10. “A tangent is always above the curve.”

Not true — depending on curvature, tangent may lie **below** the curve locally.